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Home Sports Pro Kabaddi League 2019 Playoffs/Eliminator: PKL Season 7 Schedule, Final, Timings, Venue,...
Pro Kabaddi League 2019 Playoffs/Eliminator: PKL Season 7 Schedule, Final, Timings, Venue, Teams and Prize Money
Pro Kabaddi League (PKL) 2019 Playoffs/Eliminator: Schedule, Final, Timings, Venue, Teams and Prize Money
Pro Kabaddi League 2019 playoff is scheduled to start on October 14. This week is considered to be the toughest of PKL season 7. Both the games will be played at the EKA Arena by TransStadia, Ahmedabad. The organizers have chosen Ahmadabad as the venue/host city to elevate the fan experience.
The iconic stadium will see the two eliminator, semi-finals, the PKL 2019 final matches. The grand finale is all set to happen on October 19, Saturday. The six teams include Dabang Delhi, Bengal Warriors, Haryana Steelers, UP Yoddha, U Mumba, and Bengaluru Bulls. DD were the first to qualify for the playoffs.
After that came BW, HS, UPY, UM, and BB. This season of Pro Kabaddi has arrived with a consolidated prize money of Rs 8 crores. The winners of PKL 2019 will receive Rs 3 crore as the champion's prize money. The runners-up will get Rs 1.8 crores, while the second and third runner-ups will get Rs 90 lakh each, followed by Rs 45 lakh each for fifth and sixth placed teams.
There will be two matches per day until the final one. In the start of this season, there were a total of 12 teams which participated in the competition for the title. They are as follows: Bengal Warriors, Bengaluru Bulls, Dabang Delhi KC, Gujarat Fortunegiants, Haryana Steelers, Jaipur Pink Panthers, Patna Pirates, Puneri Paltan, Tamil Thalaivas, Telugu Titans, U Mumba and UP Yoddha.
You can watch the matches on TV channels – Star Sports 1 and Star Sports 1 HD. If you want to stream it online, then you can go for the Hotstar option. Hotstar will make the live stream available for the fans and its subscribers of both Android/iOS apps and web platform.
Pro Kabaddi League 2019 Play-off Fixtures: PKL Season 7 Schedule
1 Pro Kabaddi League 2019 Play-off Fixtures: PKL Season 7 Schedule
2 PKL 2019 Season 7 Semi-finals Fixtures:
3 Pro Kabaddi League 2019 Final
Monday, October 14 – The match will start at 7:30 pm, and UP Yoddha will face the reigning champions Bengaluru Bulls.
Monday, October 14 – The second eliminator will see U Mumba clashing with Haryana Steelers at 8:30 pm.
PKL 2019 Season 7 Semi-finals Fixtures:
Wednesday, October 16: The first semi-finale will be played between Dabang Delhi and the winner of the first eliminator. The game will start at 7:30 pm.
Wednesday, October 16: The second semi-finale will start at 8:30 pm. It will be played between Bengal Warriors and the winner of the second eliminator.
Pro Kabaddi League 2019 Final
The final match of the PKL season 7 will be played at 7:30 pm on October 19, 2019.
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Hoping for some tips on how to get around a problem. I'm using a heat press to flatten fibre prints - there are sixty of them, 16x20 inches, Ilford MGWT. The prints were air dried and are reasonably curly so we want to flatten them. Using correct procedure with the heat press, most of them are coming out fine. However there are about fifteen prints that are quite wavy along the short edges. When we put these through the press they came out with obvious cracks along those edges. We stopped after this happened to two prints, but now have the dilemma of how to proceed. The only thing I can think of is to rewash the prints until hopefully they lose the wave, then dry them and hope they end up in a state that enables them to go through the press OK. I'm just wondering if anyone else has encountered this scenario and if there are any known solutions. Thanks.
I would rewash them and dry them slowly. Prints tend to curl if they are dried too fast, they emulsion side tends to dry quicker hence the curl. You could also dry them pegged back to back on a clothesline. Are you using drying racks?
You might try some steam from a tea kettle on the back of the print to relax the paper in the curly edges. Then stick back in the press when the moisture from the steam is dry enough to not cause problems in the press.
You will need to raise the humidity in the mounting room as this is a known issue with low humidity conditions... I hate mounting hot prints in the winter as its really dry , so we pump up to about 50% with a portable room humidifier and it does the trick.
It is pretty humid here in Sydney, but the prints are very dry, having been made last December.
They were dried on racks, face up, which is how I will dry them again.
Can't use pegs as the image goes right to the edge of the paper, so can't risk any marks.
I will also test one with steam.
The key to flattening large prints is to do it slowly, in stages. You can start by placing a board (with a lyer of paper on them as cushion & 1/2" (13mm) plywood, 1/4" (6mm) masonite, etc.) over them and slowly add a couple of books after a day or two. Then place them in the press for a few minutes with a light pressure. Remove them and let them cool face down on a clean surface under the same board you used previously. Then repeat the process and after about three or four sessions in the press with light pressure do one final run at full pressure. Another option would be to re-wet them with a flattening agent which will help to stabilize them.
You may want to flatten them soon after the have dried or place them under some weight until you can get to them.
I have had a similar experience. Leaving a generous white border also helps - trim 2cm all around after flattening.
The test print that I rewashed on Saturday had dried quite flat when I returned on Monday. Flattened easily in the heat press, so that's good news that a wash can get rid of the waviness. This afternoon I have rewashed the remaining wavy prints and will check them tomorrow.
Yeah, in retrospect we wished that we had bought 20x24 and then trimmed to 16x20, but we didn't anticipate this problem.
Another thing that is interesting is that the same paper is drying quite flat in the current hot and humid weather, but dried really wavy a month ago. It must have been a drier heat then.
I have very little experience with heat presses so I'm learning a lot of new stuff.
"They were dried on racks, face up"
I was taught to dry fiber prints face down. Since the paper wants to curl to the emulsion side the weight of the paper reduces the curl. It makes a big difference. Of course you want to keep your screens clean.
I've noticed that too. Where I live we have strong seasons - prints always dry much flatter in summer when the humidity is higher, and tend to have a lot of waviness during winter when the central heating is on and the indoor humidity is extremely low. I used to take the drying screens into the bedroom where we run a humidifier in winter, but this risks prints getting stepped on in the dark.
What works for me, and is better for marital relations, is "dummy" prints. I use Kostiner screens, which are stackable, so in winter I usually put the good prints on the lower screen(s) and some throwaways (fully washed, of course) on the top-most screen. This slows down the drying rate of the prints lower down and reduces their curl.
I also dry my prints face-down so they curve less, as Don Mishler describes. I usually wipe the screens with a damp cloth immediately before use to remove dust that might otherwise decide to attach itself to the image, and the screens get an occasional gentle cleaning in the shower as well.
Drying the prints face up may be the issue. It will add to the curl of the paper greatly. Control humidity in the room to about 50%; both when drying the print and after removing the prints from the dry mount press after mounting. After mounting put them under a weight both to allow the glue to solidify and for humidity equilibration. If you are printing to the edge of the paper how are you trimming the mounting tissue?? The problem may be that the tissue is not to the edge. Also you may get cracks at the edge because the emulsion is more fragile on the edge.
THIS WORKS. I DO THIS.
I had the same problem. I live in midwest. Summer - humid. Winter - dry.
I was air drying on screens. Face down. was ok.
have carpet in work room.
prints face down on screens.
cover with 2 layers of towels. maybe 1 layer might work.
have a piece of mat board in press.
put print in between a piece of paper and mat board.
it should start to flatten now.
flip it over, press again.
you can stack 10 or so prints. let cool. a few hours under weight.
Unfortunately this is being done in a public darkroom where we have no control over humidity or other aspects. I don't trust the screens to be clean, hence drying face up to avoid other problems. We are not using the heat press to dry mount, just to flatten the prints so they can be bonded to aluminium at a framing place. I think they use adhesive rather than heat. The aim is to deliver them prints as flat as possible.
It's like a sauna in Sydney today, hopefully that helps the non-wavy drying. It's also a bit of a big job because there are sixty to deal with.
You are asking for a lot of whoopass if your framer is cold mounting fibre prints with adhesive rather than hot mounting... just sayin.


He wanted to top me, and I was ok with that, but told him he needed to use a condom. It was all good… several positions, etc. Then at some point, I felt around and realized he had pulled off the condom, and was barebacking me! BLACKLISTED FOREVER.


text:
KGS – Kyrgyzstani Som
Country: Kyrgyzstan
Code: KGS
Symbol: none
The som is the official currency of the Kyrgyz Republic, also known as Kyrgyzstan, a former republic of the Soviet Union. The som is subdivided into 100 tyiyn. The som was introduced in 1993, shortly after the breakup of the Soviet Union and the reestablishment of Kyrgyzstani independence. The name of the currency derives from the traditional word for currency in the Kyrgyz language, which translates into English as "pure." Even when the Soviet ruble was the official currency, it was referred to locally as som, and the word was written on the back of ruble banknotes. The current Kyrgyzstani som is issued by the National Bank of the Kyrgyz Republic.
The Kyrgyzstani som was first introduced in 1993 at a rate of 1 som: 200 Soviet rubles. The currency was initially issued directly by the government, but that duty shifted to the National Bank of the Kyrgyz Republic when it was established on 29 July 1997. Kyrgyzstan's central bank was formed with the primary duty of achieving and maintaining price stability. However, although the bank retains the right to implement monetary policy, it rarely does so, instead deferring to the economic policy of the government. Another function rarely performed by the bank is to regulate and supervise credit. Kyrgyzstani credit institutions are rarely used because of high interest rates and the traditional use of cash for most every type of transaction. The national bank also maintains foreign reserves, but as of 2004, foreign currency and gold reserves amounted to only A$459 million.
Kyrgyzstani som is circulated through a series of coins and banknotes. Coins are minted by the Kazakhstan mint in denominations of 10 and 50 tyiyn and 1, 3, 5 and 10 som. Banknotes are now in their third series and are printed in denominations of 1, 5, 10, 20, 50, 100, 200, 500, 1000 and 5000 som.
Kyrgyzstan is the second-poorest nation of Central Asia, next to Tajikistan. The average wage in the country is A$0.78 per hour and 40 percent of the population lives below the poverty line. In the 1990s, after the Soviet breakup, the GDP of the country dropped severely, but regained some footing in the last decade. However, the GDP dropped again by 3 percent in 2010. The primary industry of Kyrgyzstan is agriculture, followed by mining and textile manufacturing. The outlook of the Kyrgyzstani economy remains unfavourable and holding Kyrgyzstani som on the foreign exchange is considered very risky.

redpajama_set_name: RedPajamaCommonCrawl


Think you know the sound of Atlantic Records in the 60s? Think again – because this stunning series opens the vault to a huge amount of rare treasures from the legendary label! The package is amazing – and offers up rare tracks that were mostly only issued by Atlantic on singles – including some on the other imprints associated with the company – some by names we never even knew recorded for the label, others by bigger names, but represented here by obscure material that goes way past their hits – and which is ultra cool because of that. There's also a fair bit more deep soul than you might expect from the usual Atlantic mix – tremendous vocals, and often in less hit-heavy modes – which makes the whole thing a great revelation – and just one more reason to bow down and give even more expect to one of the key companies who helped soul music find a home in the 60s. Plus – as if that's not enough – the whole thing comes in a very cool 7" gatefold package – which emphasizes the from-singles nature of the collection. Volume 5 is overflowing with goodies – titles that include "Soul Power" and "Sly Girl" by Derek Martin, "Tryin To Reach My Goal" and "Sufferin City" by Johnny Copeland, "What's Going To Happen To Me" by Jeanne & The Darlings, "Thank You Baby For Loving Me" and "I Can't Live Without You" by Soul Brothers Six, "That Kind Of Lovin" and "She's About A Mover" by Otis Clay, "The One You Can't Have All By Yourself" by Shirley Walton, "Can't Find No Happiness" and "A Great Big Thing" by Barbara Brown, "Somebody Needs You" and "Baby What You Got" by Darrell Banks, "Please Operator" by Tony & Tyrone, "Ain't Got No Girl" by William Bell, "Let Me Down Easy" by Alvin Robinson, "Whatcha Gonna Do About It" by Ben E King & Dee Dee Sharp. "Don't Get Caught" by Mable John, "Here I Am" by Linda Lyndell, "It's You For Me" by The Amazers, and "Need Your Love So Bad" by Solomon Burke. © 1996-2019, Dusty Groove, Inc.


text:
Immersing Into Virtual Reality: DigiPen (Singapore) Provides VR Elective
The meteoric rise of virtual reality (VR) has seen many companies scrambling to take advantage of the new technology — from major hardware developers to small, independent game studios and more.
Earlier this year, Sony released their VR headset for the PlayStation 4 in October, bringing a blend of immersive VR games directly to home consoles. The indie game studio Gattai Games, founded by a group of DigiPen (Singapore) alumni, is working to bring their horror game Stifled to VR platforms in 2017, and there has been a rise of other Singapore game startups that are specializing in VR, including IgniteVR, Kaiju Den, and Actually Sane Studios.
While the market for VR is only beginning to emerge, the demands for VR content is high, especially in China where arcades dedicated to providing virtual experiences to their consumers are rising in numbers. These demands are expected to increase in the coming years as the technology expands to address the needs of consumers outside of the digital entertainment market. As such, DigiPen (Singapore) is working to prepare students to enter the VR industry through the GAT401 elective.
The students in this class are tasked to create at least one project for the HTC Vive or the Oculus Rift. These two devices are dedicated VR headsets that were designed to fully immerse the player into a virtual world. Their main difference is that the Vive is a room-based VR system that tracks a user's movement through a small physical area using sensors and specialized controllers, whereas the Oculus is a sit-down VR system that does not require a large space to experience a game.
Joe McGinn, Department Chair of Game Software Design and Production and the lecturer for the GAT401 elective, says he challenges students to approach their projects as if they were developing a professional product. One such benchmark for commercial VR titles, he said, is that the game has to remain above 90 frames per second (FPS) at all times. "One of the reasons why we have to keep the game running above 90 FPS is because people will get motion sickness from the lag," he explained. "When the player's movement does not match what the eye sees, the brain interprets that as the body being poisoned, and that's why players will feel dizzy."
In addition to creating their VR prototypes, students in the GAT401 elective also get to hear from invited guest speakers who come to class and talk about their involvement in the industry — as well as their predictions about where they think VR is headed. Representatives from one of the invited companies, IgniteVR, recently shared their insights into the production of their first game, 9 Grids, describing how the introduction of the HTC Vive had given them the boost they needed to launch on Steam. They even stayed back at the end of the class to take questions and give advice to the students who are looking to develop VR games.
Roy Koo, the founder of IgniteVR, shared his excitement about DigiPen (Singapore)'s efforts in being at the forefront of VR development. "It's great that the students are pushed to create game projects that are up to industry standards," said Roy, who saw some students' prototypes during the DigiPen (Singapore) Career Fair. "There are bound to be really original ideas that they'll come up with, and it's good that they are exploring the possibilities of VR through this course."
DigiPen (Singapore) also partnered with HTC Vive to host their VR Bootcamp event last November. The event was open to the public, so VR developers and enthusiasts were able to learn what it takes to develop games and simulations fitted for VR. In addition to other presenters — including professor McGinn, who shared about his own experience in working with VR — developers from NVidia and Unity spoke about the tools they are providing to help other VR developers with their design processes.
Games aren't the only future application for VR technology. Those interested in pursuing VR development after GAT401 will see demands rising in other types of interactive experiences, such as serious and virtual training exercises in fields like medicine, national defense, and more.
PMETs Successfully Converted Into AI Software Developers Through Attach-and-Train Program
DigiPen (Singapore) Launches New Attach-and-Train (AnT) Program for Embedded Software Developer
DigiPen (Singapore) Joins as Official Partner of National Data Science Challenge, Organized by Shopee


Stand-out between the top two picks. DREAM KISSES has two placings from three runs this prep and gets the blinkers back on, among the main chances. MASTERATI draws to do no work, must be considered. FEMME FIREBALL resumes from a 19 week spell and won both trials, still in this. SCORCHA placed last start at long odds at Sunshine Coast when first up and generally races near the speed, in with a chance.
Femme Fireball won by 1.25L.
Placed in first start this prep. Expected to be right on the speed and won three of six as a favourite. Not out of it.
Kept chasing and just missed last start at Doomben but concern is the wide barrier. Should run fitter for two previous outings. Seems unlikely.
Ran two lengths back from the winner last start at Doomben and should run fitter for past attempts. Goes up in weight. Not here.
Won once this prep at Goulburn five runs back. Has a lot of early speed and draws to do no work. Well placed.
Back after 27 week break and placed when trialling at Sunshine Coast. Would look a lot better with the track drier. Not in this.
Faded to finish three lengths off the winner last start at Sunshine Coast and gets the blinkers back on. Has placed in two runs from three this campaign. Strong hope.
First-up after 19 week spell and won both trials. Significantly down in grade. One of the main hopes.
Placed when unwanted in the betting last start at Sunshine Coast when first up. Has a lot of early speed and up in distance. Expect to be right up there.
Finished fourth last start at Gold Coast and at home in the wet conditions. Should race on the speed. Look to others.
Faded to finish seventh last start at Doomben and returns to shorter trip. Expected to settle just off the speed. Not in this.
Narrowly beaten when heavly backed last start at Ipswich and drops 4.5kg from last run. Won once this prep at Sunshine Coast four runs back. One of the main hopes.
Just missed when heavily backed last start at Sunshine Coast. Should race on the speed and has five placings from seven runs this prep but unlikely.


ALS/MND Support Group Forums > ALS and MND Support Group > Do I Have ALS? Is This ALS? > Time to see a specialist?
Interest: UMN symptoms-no one knows wth is going on.
Tried to post before and it went to moderation for some reason. I sound further along in symptoms than some of you and I haven't even been officially diagnosed yet.
2 uncles on my mother's side, her 2 brothers died of ALS. I started having symptoms in my early 20's though I didn't realize it. Only recently heard of an early onset form of ALS. Started with a limp on my right side, weak ankles, foot drop. Went from a cane to a walker to a scooter in 2005. Really bad leg spasms. Now I can't walk and can only stand a short time with bilateral support because of my balance. What balance? The progression is so slow. Grateful for that.
I only recently thought of seeing a specialist because I noticed my arms are getting fatigued with use. No neurologist has been able (or willing?) to diagnose me. I've been on a steady decline for 13 years now. I'm so tired all the time. But there's no noticeable atrophy. I don't know what to expect next.
Now the ALS specialist who treated my uncle in San Antonio wants me to make an appointment. When do I need to take the leap? How do I know it's appropriate to go to a different state to see this doctor? Or maybe I'm just scared to make the appointment.


text:
\section{Introduction}

Let $X$ be a nonsingular toric variety over $\mathbb C$ under the action of the algebraic torus $T$. A vector bundle $\pi : \mathcal{E} \rightarrow X$ is said to be a toric vector bundle on $X$ if it has an equivariant $T$-structure, i.e. an action of the torus on $\mathcal{E}$ which is linear on fibres and the morphism $\pi$ is $T$-equivariant. Toric vector bundles were first classified by Kaneyama in \cite{kane}. In \cite{kly}, Klyachko gave a description of toric vector bundles in terms of finite dimensional vector spaces with the family of decreasing $\mathbb Z$-graded filtrations which satisfy a compatibility condition. Tangent bundles of nonsingular toric varieties arise as natural examples of toric vector bundles. Following \cite{ehlers}, the tangent bundle of any nonsingular toric variety has a natural topological decomposition into sum of line bundles, each of which arises from a one-dimensional cone of the associated fan. This result is not true in case of equivariant algebraic splitting. In Proposition \ref{18} we characterize all nonsingular toric varieties whose tangent bundle splits equivariantly into direct sum of line bundles.

A Bott tower of height $n$ $$M_n \rightarrow M_{n-1} \rightarrow \cdots \rightarrow M_2 \rightarrow M_1 \rightarrow M_0=\{ \text{point} \}$$ is defined inductively as an iterated projective bundle so that each stage $M_k$ of the tower is of the form $\mathbb{P}(\mathcal{O}_{M_{k-1}}  \oplus \mathcal{L}$) for an arbitrarily chosen line bundle $\mathcal{L}$ over the previous stage $M_{k-1}$. Bott towers were shown to be deformation of Bott-Samelson varieties by Grossberg and Karshon in \cite{grossbergkarshon}. The Bott tower of height $n$ is completely determined by $\frac{n(n-1)}{2}$ integers $\{c_{i, j}\}_{\{1\leq i < j \leq n \}}$, known as the Bott numbers and these are arranged in an $n \times n$ upper triangular matrix with $1$'s in the diagonal, called the Bott matrix. Without loss of generality, the entries of the Bott matrix can be assumed to be non-negative (Theorem \ref{12}). In Proposition \ref{15} we show that the tangent bundle of $M_k$ splits equivariantly if and only if the Bott matrix is the identity matrix.

Let $\mathcal{L}$ be a line bundle on a projective variety $X$. Then $\mathcal{L}$ is said to be very ample if the rational map  $X \dashrightarrow \mathbb{P}H^0(X, \mathcal{L})$, determined by its complete linear system $|\mathcal{L}|$, is everywhere defined morphism which becomes an embedding. Generalizing this, the notion of higher-order embeddings were introduced by Beltrimetti and Sommese in \cite{rider}, among which the notion of $s$-jet ampleness is the strongest. In Theorem \ref{16}, we give a criterion for $s$-jet ampleness of line bundles on Bott towers using Bott numbers.

A nonsingular projective algebraic variety is said to be Fano (respectively, weak Fano) if its anticanonical divisor is ample (respectively, nef and big). We give ampleness and nefness criterion for line bundles on Bott towers\footnote{While working on this project, it came to our notice that Narasimha B. Chary has obtained similar criterion for ample and nef line bundles on Bott towers in \cite{bonala2017mori} using a different method.} in Corollary \ref{2} and characterize nef and big line bundles on Bott towers in Theorem \ref{5}, hence in particular, give a characterization of Fano (respectively, weak Fano) Bott towers.

If a vector bundle $\mathcal{E}$ is ample, then it is known that the determinant bundle $\text{det}(\mathcal{E})$ is also ample, the converse need not be true in general. But if $\mathcal{E}$ is semi-stable vector bundle on a smooth projective curve over algebraically closed field, the converse holds (see  \cite[Theorem $3.2.7$]{huybrechts2010geometry}). In this paper we prove that the converse holds for discriminant zero semi-stable toric vector bundles on nonsingular projective toric variety in Proposition \ref{7}, using the results of Hering, Musta{\c{t}}{\u{a}} and Payne \cite{hering2010positivity}. We characterize semi-stable bundles on Hirzebruch surface in Proposition \ref{9}.

\noindent
{\bf Acknowledgements:} We thank our advisors Arijit Dey and V. Uma for their valuable guidance throughout this project. We also thank D.S. Nagaraj for his valuable comments and suggestions on an earlier version of the manuscript. Finally we thank Council of Scientific and Industrial Research (CSIR) for their financial support.

\section{Some basic results}

\subsection{Toric vector bundle}

Let $T \cong \left(\mathbb C^*\right) ^n$ be an algebraic torus. Let $M=\text{Hom}(T, \mathbb C^*) \cong \mathbb Z^n$ be its character lattice and $N=\text{Hom}_{\mathbb Z}(M, \mathbb Z)$ be the dual lattice. Let $\Delta$ be a fan in $N_{\mathbb R}:=N \otimes_{\mathbb Z} \mathbb R$ which defines a toric variety $X=X(\Delta)$ under the action of the torus $T$. Let $\Delta(1)$ denote the edges of $\Delta$ and $\sigma(1)$ denote the edges of a cone $\sigma$ in $\Delta$. For each $\rho \in \Delta(1)$, let $D_{\rho}$ denote the $T$-invariant prime divisor corresponding to $\rho$. \\
A vector bundle $\pi : \mathcal{E} \rightarrow X$ is a toric or $T$-equivariant vector bundle on $X$ if $\mathcal{E}$ has a lift of the action of $T$ such that:
\begin{enumerate}
\item The action of $T$ on $\mathcal{E}$ is linear on fibres of $\pi$.
\item $\pi$ is $T$-equivariant i.e., for all $ e \in \mathcal{E}$ and $t \in T$, $\pi(t \cdot e) = t \cdot \pi(e)$.
\end{enumerate}

\noindent
It is known that any line bundle on toric variety is isomorphic to a toric line bundle. In fact, any line bundle on $X$ is isomorphic to $\mathcal{O}_X(D)$ for some $T$-invariant Cartier divisor $D$, which determines the equivariant structure of the line bundle. The tangent bundle and cotangent bundle of a nonsingular toric variety are natural examples of toric vector bundles.

\noindent
Toric vector bundles were first classified by Kaneyama in \cite{kane}. Later Klyachko gave a classification in terms of filtrations of certain vector space as follows:

\begin{thm}\label{1} \cite[Theorem $2.2.1$]{kly} The category of toric vector bundles over the toric variety $X=X(\Delta)$ is equivalent to the category of vector spaces $E$, with a family of decreasing $\mathbb Z$-filtrations $E^{\rho}(i)$, for $\rho \in \Delta(1)$, which satisfy the following compatibility condition:\\
$({\bf C})$ for any $\sigma \in \Delta$, there exists a $\widehat{T}_{\sigma}$-grading $E=\oplus_{\chi \in \widehat{T}_{\sigma}}E^{[\sigma]}(\chi)$, for which
$$E^{\rho}(i)=\sum_{\langle \chi, v_{\rho} \rangle \geqslant i}E^{[\sigma]}(\chi)$$ for all $\rho \in \sigma(1)$, where $v_{\rho}$ denotes the primitive ray generator of the ray $\rho$.
\end{thm}

Recall that a family of linear subspaces $\{V_\lambda \}_{\lambda \in \Lambda}$ of a finite dimensional vector space $V$ is said to form a distributive lattice if, there exists a basis $B$ of $V$ such that $B \cap V_\lambda$ is a basis of $V_\lambda$ for every $\lambda \in \Lambda$. If $X$ is nonsingular, then condition $({\bf C})$  of Theorem \ref{1} is equivalent to the following: for each $\sigma \in \Delta$, the collection of subspaces $\{E^{\rho}(i) \}_{\rho \in \sigma(1), i \in \mathbb Z}$ of $E$ forms a distributive lattice (See \cite[Remark $2.2.2$]{kly}). To see this, note that if the condition $({\bf C})$ is satisfied, for each $\sigma \in \Delta$, the eigen basis of the $\widehat{T}_{\sigma}$-grading $E=\oplus_{\chi \in \widehat{T}_{\sigma}}E^{[\sigma]}(\chi)$ serves as the required basis. Conversely, for $\sigma \in \Delta$, let $B_{\sigma}=\{e_1, \ldots, e_n \}$ be a basis of $E$ such that $E^{\rho}(i) \cap B_{\sigma}$ is a basis for $E^{\rho}(i)$. Then for each $e_j$ there exists an integer $n^j_{\rho}$ such that $e_j \in E^{\rho}(i) \cap B_{\sigma}$ for $i \leq n^j_{\rho}$ and $e_j \notin E^{\rho}(i) \cap B_{\sigma}$ for $i > n^j_{\rho}$. Since $\sigma$ is nonsingular cone, define  characters of $T_{\sigma}$ by $\chi_j:N_{\sigma} \rightarrow \mathbb Z$, $v_{\rho} \mapsto n^j_{\rho}$ for $\rho \in \sigma(1)$, which yields the required $\widehat{T}_{\sigma}$-grading $E=\oplus_{j=1}^n \text{Span}(e_j)$.

\begin{ex}[Filtrations for line bundles]{\rm
Let $\mathcal{L}=\mathcal{O}_X(D)$ be a toric line bundle on $X$ for some $T$-invariant Cartier divisor $D=\sum_{\rho \in \Delta(1)}a_{\rho}D_{\rho}$, $a_{\rho} \in \mathbb Z$. Then the associated filtrations $(L, \{L^{\rho}(i)\}_{\rho \in \Delta(1)})$ are given by:
\[ L^{\rho}(i) = \left\{ \begin{array}
{r@{\quad \quad}l}
\mathbb C & i \leqslant a_{\rho} \\

 0 & i > a_{\rho}
\end{array} \right. \]

\noindent
Note that the filtrations define a function $f: \Delta(1) \rightarrow \mathbb Z$, given by $\rho \mapsto a_{\rho}$. Conversely when $X$ is nonsingular, any such function $f$ defines a toric line bundle whose associated filtrations are given by:
\[ L^{\rho}(i) = \left\{ \begin{array}
{r@{\quad \quad}l}
\mathbb C & i \leqslant f(\rho) \\

 0 & i >  f(\rho)
\end{array} \right. \]
}
\end{ex}

Note that for any two toric vector bundles $\mathcal{E}$ and $\mathcal{F}$ with the associated filtrations $(E, \{ E^{\rho}(i) \}_{\rho \in \Delta(1)})$ and $(F, \{ F^{\rho}(i) \}_{\rho \in \Delta(1)})$ respectively, the associated filtrations for $\mathcal{E \oplus F}$ and  $\mathcal{E \otimes F}$ are given by $(E \oplus F, \{E^{\rho}(i) \oplus F^{\rho}(i)\}_{\rho \in \Delta(1)})$ and $(E \otimes F, \{\sum_{s+ t=i} E^{\rho}(s) \otimes F^{\rho}(t) \}_{\rho \in \Delta(1)})$ respectively. In particular, if $\mathcal{F}$ is a toric line bundle $\mathcal{O}_X(D)$, for some $T$-invariant Cartier divisor $D=\sum_{\rho \in \Delta(1)}a_{\rho}D_{\rho}$ with the associated  filtrations $(L, \{ L^{\rho}(i) \}_{\rho \in \Delta(1)})$ then the associated filtrations for $\mathcal{E} \otimes \mathcal{O}_X(D)$ is given by $(E , \{ E^{\rho}(i-a_{\rho}) \}_{\rho \in \Delta(1)} )$.  We say that a toric vector bundle splits if it is equivariantly isomorphic to a direct sum of toric line bundles.

\begin{ex}[Filtrations for tangent bundles]\label{10}{\rm
Let ${T}_X$ be the tangent bundle on the nonsingular toric variety $X=X(\Delta)$. Then the associated filtrations $(E, \{E^{\rho}(i)\}_{\rho \in \Delta(1)})$ are given by:\\
\[ E^{\rho}(i) = \left\{ \begin{array}
{r@{\quad \quad}l}
N_{\mathbb C} & i \leqslant 0 \\
\text{Span }( v_{\rho} ) & i=1 \\
 0 & i > 1
\end{array} \right. \]
}
\end{ex}

Let $X=X(\Delta)$ be a nonsingular toric variety and $\pi : \mathcal{E} \rightarrow X$ be a toric vector bundle on $X$ with associated filtrations $\left( E, \{E^{\rho}(i) \}_{\rho \in \Delta(1)} \right)$. Let $\pi \mid_{\mathcal{F}} : \mathcal{F} \rightarrow X$ be a toric subbundle with associated filtrations $\left( F, \{F^{\rho}(i) \}_{\rho \in \Delta(1)} \right)$. We characterize the filtration data for $\mathcal{F}$ as follows:

\begin{prop}\label{6}
$\mathcal{F}$ is a toric subbundle of $\mathcal{E}$ if and only if $F \subset E, \ F^{\rho}(i)= E^{\rho}(i) \cap F$ and for each $\sigma \in \Delta$, there exists a basis $B^F_{\sigma}$ of $F$ which extends to a basis $B^E_{\sigma}$ of $E$ such that $F^{\rho}(i) \cap B^F_{\sigma}$ forms a basis of $F^{\rho}(i)$ and $E^{\rho}(i) \cap B^E_{\sigma}$ forms a basis of $E^{\rho}(i)$ for all $\rho \in \sigma(1)$.
\end{prop}

\noindent
{\bf Proof:} Fix $\sigma \in \Delta$, then $F$ is a $T_{\sigma}$ stable subspace of $E$. Since $T_{\sigma}$ is diagonalizable, we can write $E=F \oplus F'$, where $F'$ is the $T_{\sigma}$ stable complement of $F$ in $E$. Let $\{ f_1, \ldots, f_r \}$ be an eigen basis of $F$ (i.e, $t \cdot f_i =\chi_i(t) f_i$ for all $t \in T_{\sigma}$) which extends to an eigen basis of $E$. Hence we can write $E=\text{Span}\{ f_1, \ldots, f_r, f'_1, \ldots, f'_s \}\left(=\bigoplus_{\chi \in T_{\sigma} }E_{\chi}, \text{say} \right)$,  such that $t \cdot f'_j =\chi'_j(t) f'_j$ for all $t \in T_{\sigma}$. Note that $\chi_i, \ \chi'_j$'s need not be distinct. Also by compatibility, $E^{\rho}(i)=\bigoplus_{\langle \chi , v_{\rho} \rangle \geq i} E_{\chi}$. Here $E_{\chi}=\{f \in E : t \cdot f = \chi(t) f \ \forall \ t \in  T_{\sigma} \}=\text{Span}\{f_i : t \cdot f_i =\chi(t) f_i \ \forall \ t \in  T_{\sigma} \}+ \text{Span}\{f'_j : t \cdot f'_j =\chi(t) f'_j \ \forall \ t \in  T_{\sigma} \}$. We see that $F^{\rho}(i)=E^{\rho}(i) \cap F$. Then the choice of the bases $B^F_{\sigma}=\{ f_1, \ldots, f_r \}$ and $B^E_{\sigma}=\{ f_1, \ldots, f_r, f'_1, \ldots, f'_s \}$ works.

For the other direction, we only need to show that for each cone $\sigma$, the $T_{\sigma}$ action on $\mathcal{F}$ is induced from the $T_{\sigma}$ action on $\mathcal{E}$. Let us call $B^F_{\sigma}=\{ f_1, \ldots, f_r \}$ and $B^E_{\sigma}=\{ f_1, \ldots, f_r, f'_1, \ldots, f'_s \}$, where $B^F_{\sigma}, \ B^E_{\sigma}$ are as in the hypothesis. Fix $\rho \in \sigma(1)$, since $E^{\rho}(i)$ is decreasing full filtration, for each $f_j$ there is an integer $n^j_{\rho}$ such that $f_j \in E^{\rho}(i) \cap B^E_{\sigma}$ for $i \leq n^j_{\rho}$ and $f_j \notin E^{\rho}(i) \cap B^E_{\sigma}$ for $i > n^j_{\rho}$. Similarly, for each $f'_j$, we have $m_{\rho}^j$. Now define the characters of $T_{\sigma}$ as $\chi_j: N_{\sigma} \rightarrow \mathbb Z$ which sends $\rho$ to $ n^j_{\rho}$, and $\chi'_j: N_{\sigma} \rightarrow \mathbb Z$ which sends $\rho$ to $ m^j_{\rho}$. Then we can define the $T_{\sigma}$ action on $E$ by $$t \cdot f_j=\chi_j(t)f_j \text{ and } t \cdot f'_j=\chi'_j(t)f'_j$$ Set $E_{{\chi}_j}=$ Span $f_j$ and $E_{{\chi'}_j}=$ Span $f'_j$.
We see that with respect to this $E$ has a $\widehat{T}_{\sigma}$ grading $E=\sum E_{{\chi}_j} + \sum E_{{\chi'}_j}$ such that $E^{\rho}(i)=\bigoplus_{\langle \chi , v_{\rho} \rangle \geq i} E_{\chi}$. Now, by construction, we see that $F$ has a $\widehat{T}_{\sigma}$ grading $F=\sum E_{\chi_j}$ such that $F^{\rho}(i)=\bigoplus_{\langle \chi , v_{\rho} \rangle \geq i} F_{\chi}$, which concludes the proof.
$\hfill\square$

\begin{rmk}{\rm
Let $X$, $\mathcal{E}$ and $\mathcal{F}$ be as above and $\mathcal{F}$ be a toric subbundle of $\mathcal{E}$. Now define a filtration on $\frac{E}{F}$ by setting $\left( \frac{E}{F} \right)^{\rho}(i):=\frac{E^{\rho}(i)}{F^{\rho}(i)}$, then  $B_{\sigma}=B^E_{\sigma} \setminus B^F_{\sigma}$ serves as the required basis of $\frac{E}{F}$ which satisfies $(\bf{C})$ of Theorem \ref{1}. Then we have a short exact sequence in the category of filtered vector space with compatible filtrations: \begin{center}
$\ses{F}{E}{\frac{E}{F}}{\imath}{\pi}$ and
\end{center}
\begin{center}
$\ses{F^{\rho}(i)}{E^{\rho}(i)}{\frac{E^{\rho}(i)}{F^{\rho}(i)}}{\imath}{\pi}$ for all $\rho \in \sigma(1)$ and $i \in \mathbb Z$.
\end{center}

\noindent
So these induces the following short exact sequence in the category of toric vector bundles, \begin{center}
 $0 \rightarrow \mathcal{F} \rightarrow \mathcal{E} \rightarrow \mathcal{Q} \rightarrow 0$.
\end{center}
Hence quotient of $\mathcal{E}$ by $\mathcal{F}$ exists in the category of toric vector bundles, which is surprising since this fact is not true in general varieties.
}
\end{rmk}

It is known that the tangent bundle of any nonsingular toric variety splits topologically into direct sum of line bundles, although this result is not true in case of equivariant algebraic splitting, for example consider the tangent bundle of $\mathbb{P}^2$. Recall that, a toric vector bundle $\mathcal{E}$ over a nonsingular toric variety $X=X(\Delta)$ splits if and only if the filtrations $E^{\rho}(i)$, $\rho \in \Delta(1)$ generate a distributive lattice (see \cite[Corollary $2.2.3$]{kly}). Using this, we obtain a splitting criterion of tangent bundle of a nonsingular toric variety:

\begin{prop}\label{18}
Let $\text{dim}(X)=n$, then ${T}_X$ splits if and only if for all $\sigma \in \Delta(n)$ and any ray $\rho \in \Delta(1) \setminus \sigma(1)$ the primitive ray generator $v_\rho$ of $\rho$ is the negative of some primitive ray generator of the cone $\sigma$.
\end{prop}

\noindent
{\bf Proof: }Suppose that ${T}_X$ splits. Then there exists a basis $B$ of $E$ such that $B \cap E^\rho(1)$ is a basis of $E^\rho(1)=\text{Span}(v_\rho)$ for all $\rho \in \Delta(1)$. So for all $\rho \in \Delta(1)$ either $v_\rho$ or $- v_\rho \in B$, which concludes the forward direction.

Conversely, fix a cone $\sigma=\text{Cone }(v_{\rho_1}, \ldots, v_{\rho_n}) \in \Delta(n)$, then $\{v_{\rho_1}, \ldots, v_{\rho_n}\} \cap E^\rho(1)$ is a basis of $E^\rho(1)$ for all $\rho \in \Delta(1)$. Hence ${T}_X$ splits.
$\hfill\square$

\begin{cor}
If $|\Delta(1)| > 2n$, then ${T}_X$ does not split.
\end{cor}

\subsection{Bott tower}

\noindent
We now briefly recall the construction of Bott tower and some basic results related to it. For more details see \cite{grossbergkarshon} and \cite{civanyusuf}. Let $e_1, \ldots, e_r$ be the standard basis for $\mathbb{R}^r$, for any $r \in \mathbb N$. Given $\frac{n(n-1)}{2}$ integers $\{c_{i, j}\}_{\{1\leq i < j \leq n \}}$, the Bott tower of height $n$
$$M_n \rightarrow M_{n-1} \rightarrow \cdots \rightarrow M_2 \rightarrow M_1 \rightarrow M_0=\{ \text{point} \}$$
is constructed recursively as follows: Let $M_0$ be a point and $\xi_0$ be the trivial line bundle on $M_0$. Define $M_1$ to be $\mathbb{P}(\mathcal{O}_{M_0} \oplus \xi_0)=\mathbb{P}^1$. Let $\Delta_1 \subset \mathbb R$ be the fan of $M_1$ consisting of rays $v^1_1=e_1, v^1_2=-e_1$. Note that $\text{Pic}(M_1)=\mathbb Z [D_2]$, where $D_2$ is the invariant prime divisor corresponding to $v^1_2$. Let $\xi_1$ be the toric line bundle on $M_1$ whose associated filtrations are given by:
\begin{center}
$
 \xi_1^{v^1_1}(l) = \left\{ \begin{array}{ccc}

\mathbb C & l \leqslant 0 \\

 0 & l > 0
\end{array} \right. \ \  \ \ \text{and \ \ \ \ }
$
$
\xi_1^{v^1_2}(l) = \left\{ \begin{array}{ccc}

\mathbb C & l \leqslant c_{1, 2} \\

 0 & l > c_{1,  2}
\end{array} \right.
$
\end{center}

\noindent
Let $M_2:=\mathbb{P}(\mathcal{O}_{M_1} \oplus \xi_1)$ be the $\mathbb{P}^1$-bundle on $M_1$, where the fibre $\mathbb{P}^1$ has ray generators $e'_1, e'_0=-e'_1$. Consider the map $p : \mathbb R \rightarrow \mathbb R \oplus \mathbb R$, given by $v^1_1 \mapsto (v^1_1, 0)$ and $v^1_2 \mapsto (v^1_2, c_{1, 2}e'_1)$. Then the ray generators of $M_2$ are $v^2_1=e_1,v^2_2=e_2,v^2_3=-e_1 + c_{1, 2}e_2, v^2_4=-e_2$. The four maximal cones are $\text{Cone }(v^2_1, v^2_2), \text{Cone }(v^2_1, v^2_4), \text{Cone }(v^2_2, v^2_3)$ and $\text{Cone }(v^2_3, v^2_4)$. Repeating this process, at the $k$-th stage, we have a $k$-dimensional nonsingular projective toric variety $M_k$, whose fan structure is as follows: Let $v_i=e_i$ for $i=1, \ldots, k$,
$v_{k+i}=-e_i+c_{i, i+1}e_{i+1}+\cdots +c_{i, k}e_k \text{ for }i=1, \ldots, k-1$
and $v_{2k}=-e_k$.\\
The fan $\Delta_k$ of $M_k$ is complete and consists of these $2k$ edges and $2^k$ maximal cones of dimension $k$ generated by these edges such that no cone contains both the edges $v_i$ and $v_{k+i}$ for $i=1, \ldots, k$.\\
Consider a line bundle $\xi_k$ on $M_k$ whose associated filtrations are given by:
\begin{center}
$
\xi_k^{v_i}(l) = \left\{ \begin{array}{ccc}

\mathbb C & l \leqslant 0 \\

 0 & l > 0
\end{array} \right. \ \
$
and \ \ \ \
$
\xi_k^{v_{k+i}}(l) = \left\{ \begin{array}{ccc}

\mathbb C & l \leqslant c_{i, k+1} \\

 0 & l > c_{i, k+1}
\end{array} \right.
$
for $i=1, \ldots, k$.
\end{center}

\noindent
Then $M_{k+1}:=\mathbb{P}(\mathcal{O}_{M_k} \oplus \xi_k)$. The integers $c_{i, j}$'s are called Bott numbers and they are arranged in a $n \times n$ upper triangular matrix with $1$'s in the diagonal, called Bott matrix.

\noindent
Let $D_i:=D_{v_i}$ denote the invariant prime divisor corresponding to an edge $v_i$. The Picard group of the Bott tower is given by $\text{Pic}(M_k)=\mathbb Z [D_{k+1}] \oplus \ldots \oplus \mathbb Z [D_{2k}]$ as
\begin{center}
$D_i \sim_{\text{lin}} D_{k+i}-c_{1, i}D_{k+1}-\cdots-c_{i-1, i}D_{k+i-1}$ for $i=1, \ldots, k$.
\end{center}
Hence any divisor $D$ on $M_k$ can be written as $D=\sum_{i=1}^{k}a_iD_{k+i}$, $a_i \in \mathbb Z$ for $i=1, \ldots, k$.\\

\noindent
The key observation regarding the Bott numbers is the following:

\begin{thm}\label{12}
The integers $\{c_{i, j}\}_{\{1\leq i < j \leq n \}}$ can be assumed to be non-negative.
\end{thm}

\noindent
{\bf Proof:} We prove it by induction on $n$.\\
For $n=2$ the corresponding integer is $c_{1, 2}$ and $M_2=\mathcal{H}_{c_{1, 2}}$, Hirzebruch surface and it is well known that $c_{1, 2}$ can be assumed to be non-negative. By induction hypothesis assume that $c_{i, j}$'s are non-negative for $1\leq i < j \leq k $.\\
Consider a line bundle $\eta_k$ on $M_k$ with associated filtrations:
\begin{center}
$
 \eta_k^{v_i}(l) = \left\{ \begin{array}{ccc}

\mathbb C & l \leqslant 0 \\

 0 & l > 0
\end{array} \right. \ \ \
$
and \ \ \
$
 \eta_k^{v_{k+i}}(l) = \left\{ \begin{array}{ccc}

\mathbb C & l \leqslant |c_{i, k+1}| \\

 0 & l > |c_{i, k+1}|
\end{array} \right.
$
for $i=1, \ldots, k$.
\end{center}

\noindent
We show that there is a line bundle $L_k$ on $M_k$ such that $(\mathcal{O}_{M_k} \oplus \xi_k)\otimes L_k \cong \mathcal{O}_{M_k} \oplus \eta_k$, where $\xi_k$ is as before. Let $S=\{v_{k+i} | c_{i, k+1} < 0 \}$ and the filtrations associated to $L_k$ be as follows:\\
If $\alpha \notin S$, $ L_k^{\alpha}(l) = \left\{ \begin{array}{ccc}

\mathbb C & l \leqslant 0 \\

 0 & l > 0
\end{array} \right. \ \
$
and if $\alpha \in S$, say $\alpha=v_{k+i}$ ,
$L_k^{\alpha}(l) = \left\{ \begin{array}{ccc}

\mathbb C & l \leqslant -c_{i, k+1}  \\

 0 & l > -c_{i, k+1}
\end{array} \right.
$ \\

\noindent
Note that the associated filtrations for $\mathcal{O}_{M_k} \oplus \eta_k$ are given by:
\begin{center}
$
(\mathcal{O}_{M_k} \oplus \eta_k)^{v_i}(l) = \left\{ \begin{array}{ccc}

\mathbb C \oplus \mathbb C & l \leqslant 0 \\

 0 & l > 0
\end{array} \right.
$
\end{center}
\begin{center}
when $\alpha=v_{k+i} \notin S $,
$(\mathcal{O}_{M_k} \oplus \eta_k)^{\alpha}(l) = \left\{ \begin{array}{ccc}

\mathbb C \oplus \mathbb C & l \leqslant 0 \\
\mathbb C    &   0 < l \leq c_{i, k+1}            \\
 0 & l > c_{i, k+1}
\end{array} \right.
$
\end{center}
\begin{center}
when $\alpha \in S$, say $\alpha=v_{k+i}$,
$ (\mathcal{O}_{M_k} \oplus \eta_k)^{\alpha}(l) = \left\{ \begin{array}{ccc}

\mathbb C \oplus \mathbb C & l \leqslant 0 \\
\mathbb C    &   0 < l \leq -c_{i, k+1}           \\
 0 & l > -c_{i, k+1}
\end{array} \right.
$

\end{center}

\noindent
By similar computations, the filtrations associated to the vector bundle $( \mathcal{O}_{M_k} \oplus \xi_k)\otimes L_k$ are same. Hence by Theorem \ref{1} these two are $T$-equivariantly isomorphic. Thus $M_{k+1}:=\mathbb{P}(\mathcal{O}_{M_k} \oplus \xi_k)$ and $\mathbb{P}(\mathcal{O}_{M_k} \oplus \eta_k)$ are isomorphic as abstract varieties, hence they are isomorphic as toric varieties also (see \cite[Theorem $4.1$]{berchtold2003lifting}).
$\hfill\square$\\

\noindent
Let $T_{M_k}$ be the tangent bundle on $M_k$ with associated filtrations $(E, \{ E^\rho(i)\}_{\rho \in \Delta_k, i \in \mathbb Z})$ (see Example \ref{10}).

\begin{prop}\label{15}
$T_{M_k}$ splits if and only if $c_{i, j}=0$ for all $ i \text{ and } j$.
\end{prop}

\noindent
{\bf Proof:} Suppose that $T_{M_k}$ splits. Then there exists a basis $B$ of $\mathbb C^k$ such that $E^{v_i}(1) \cap B$ is a basis of $E^{v_i}(1)=\text{Span }(e_i)$ for all $i=1, \ldots, k$, which implies that $B=\{ c_1e_1, \ldots,c_k e_k \}$, where $c_i$'s are $1$ or $-1$. But $E^{v_{k+i}}(1) \cap B$ has to be a basis of $E^{v_{k+i}}(1)=\text{Span }(-e_i+c_{i, i+1}e_{i+1}+ \ldots + c_{i,  k}e_k)$. This forces $c_{i, j}=0$.

Conversely, let $c_{i, j}=0$ for all $i$ and $j$. Then the choice of the basis $B= \{ e_1, \ldots, e_k\}$ of $\mathbb C^k$ works.$\hfill\square$

\begin{rmk}{\rm
In general, consider the toric line bundles $\mathcal{L}_0, \mathcal{L}_1, \ldots, \mathcal{L}_m$ over a nonsingular toric variety $X=X(\Delta)$. Then $X'= \mathbb{P}(\mathcal{L}_0 \oplus \mathcal{L}_1 \oplus \ldots \oplus \mathcal{L}_m)$ is also a nonsingular toric variety with fan $\Delta'$ given as follows: Let $\mathcal{L}_0, \mathcal{L}_1, \ldots, \mathcal{L}_m$ correspond to $\Delta$-linear support functions $h_0, h_1, \ldots, h_m$ respectively. Choose a $\mathbb Z$-basis $\{e_1, \ldots, e_m\}$ of $\mathbb R^m$ and let $e_0=-e_1- \ldots - e_m$. Consider the $\mathbb R$-linear map $\Phi: \mathbb R^n \rightarrow \mathbb R^n \oplus \mathbb R^m$ which sends $y \mapsto (y, -\sum_{j=0}^m h_j(y)e_j)$. Now let $\tilde{\sigma}_i = \text{Cone}(e_0, \ldots, \widehat{e_i}, \ldots, e_m)$ for each $0\leq i \leq m$ (here by $\widehat{e}_i$ we mean that $e_i$ is omitted from the relevant collection). Let $\tilde{\Delta}$ be the fan in $\mathbb R^m$ generated by $\tilde{\sigma}_i$ for $0\leq i \leq m$. Then $\Delta'=\{\Phi(\sigma) + \tilde{\sigma} : \sigma \in \Delta, \tilde{\sigma}\in \tilde{\Delta}\}$.

\begin{enumerate}

\item For $m \geq 2$, $T_{X'}$ does not split. To see this, consider the maximal cone $\sigma'= \Phi(\sigma) + \text{Cone}(e_0, \ldots, \widehat{e_i}, \ldots, e_m)$, where $\sigma \in \Delta$ is a maximal cone. The ray generated by $(0, e_i)$ is not in $\sigma'$ and  $(0, e_i)$ is not the negative of any of the primitive ray generators of $\sigma'$.

\item For $m=2$, if $T_{X'}$ splits then $T_X$ also splits.

\end{enumerate}
}
\end{rmk}

\subsection{s-jet ampleness}

\noindent
Let us recall the notion of $s$-jet ampleness, see \cite{rider} for details. Let $X$ be a nonsingular algebraic variety. For a point $x \in X$ let $\mathfrak{m}_x$ be the maximal ideal sheaf of $x$ in $X$.

\begin{defn}
A line bundle $\mathcal{L}$ on $X$ is said to be $s$-jet ample, if for every finite collection of points $x_1, \ldots, x_r$ the restriction
map $$H^0(X; \mathcal{L}) \longrightarrow H^0\left(X; \mathcal{L} \otimes \frac{\mathcal{O}_X}{\mathfrak{m}_{x_1}^{k_1} \otimes \cdots \otimes \mathfrak{m}_{x_r}^{k_r}} \right)$$
is surjective, where $k_1, \ldots, k_r$ are positive integers and $\sum_{i=1}^r k_i = s + 1$.
\end{defn}

Note that $\mathcal{L}$ is $0$-jet ample if and only if $\mathcal{L}$ is globally generated and $\mathcal{L}$ is $1$-jet ample if and only if $\mathcal{L}$ is very ample (see \cite[Section $2.3$]{jetample}). On nonsingular complete toric varieties the notion of ample and very ample line bundles coincide (see \cite[Theorem $6.1.16$]{Cox}). Let us recall the following generalization of Toric-Nakai criterion for $s$-jet ampleness.

\begin{thm} \cite[Proposition $3.5$, Theorem $4.2$]{DiRocco}\label{11} Let $\mathcal{L}$ be a line bundle on a nonsingular toric variety $X=X(\Delta)$. Then the following are equivalent:
\begin{itemize}
\item[(1)] $\mathcal{L}$ is $s$-jet ample.
\item[(2)] $\mathcal{L} \cdot C \geq s$, for any $T$-invariant curve C.
\end{itemize}
\end{thm}

\section{Positivity of line bundles on Bott towers}

\subsection{s-jet ampleness on Bott tower}

\noindent
We use Theorem \ref{11} to give a necessary and sufficient criterion for $s$-jet ampleness of line bundles on Bott tower in terms of the Bott numbers.

\begin{thm}\label{16}
Let $D=\sum_{i=1}^{k}a_iD_{k+i}$ be a $T$-invariant Cartier divisor on $M_k$. Then $D$ is  $s$-jet ample if and only if $a_i  \geq \ s$ for all $i=1, \ldots, k$.
\end{thm}
\noindent
{\bf Proof:} Let $D$ be $s$-jet ample. For $i=1, \ldots, k$, consider the cones $\tau_i=\text{Cone}(v_1, \ldots, \widehat{v}_i, \ldots, v_k)$. Now $D \cdot V(\tau_i)  \geq s$ for $i=1, \ldots, k$ by Theorem \ref{11}. Note that $D \cdot V(\tau_i)=a_i$ since $D_{k+l} \cdot V(\tau_i)=0$ for $l \neq i$ and $D_{k+i} \cdot V(\tau_i)=1$ (\cite[Corollary $6.3.3$]{Cox}).

Conversely, let $a_i  \geq \ s$ for all $i=1, \ldots, k$. Again  by Theorem \ref{11}, it suffices to show that $D \cdot V(\tau) \geq s$ for any $k-1$ dimensional cone $\tau \in \Delta_k$. Note that any such cone is of the form
\begin{center}
$\tau=\text{Cone}(v_1, \ldots, \widehat{v}_{i_1}, \ldots, \widehat{v}_{i_r}, \ldots, v_k,v_{k+i_1}, v_{k+i_2}, \ldots, \widehat{v}_{k+i_j}, \ldots, v_{k+i_r})$
\end{center}
for some $j=1, \ldots, r$. We have\\
$
 D_{k+m} \cdot V(\tau) = \left\{ \begin{array}{ccc}

1 & m=i_j   \\
0 & m \neq i_1, \ldots, i_r \\
\end{array} \right.  \ \
$
and \ \ \
$
  D_{k+m} \cdot V(\tau) = \left\{ \begin{array}{ccc}

 0 &   m=i_l, 1 \leq l <j ,\\
 b_{i_l} & m =i_l, j < l \leq k ,\\
\end{array} \right. \\
$

\noindent
where $b_{i_l}$'s are such that the following wall relation is satisfied:
\begin{center}
$v_{i_j}+d_1v_1+\cdots+\widehat{v}_{i_1}+\cdots+\widehat{v}_{i_r}+\cdots+d_kv_k+v_{k+i_j}+\sum_{t=j+1}^r b_{i_t}v_{k+i_t}=0$
\end{center}
where $d_1, \ldots, d_k$ are integers(\cite[Corollary $6.3.3$ and Lemma $6.3.4$]{Cox}). Then we have,
\begin{center}
$b_{i_{j+1}}=c_{i_j, i_{j+1}}$; $b_{i_{j+2}}=c_{i_{j+1}, i_{j+2}} b_{i_{j+1}}+c_{i_{j}, i_{j+2}}$;\\
  $\vdots$\\
  $b_{i_r}=c_{i_{j+1}, i_r} b_{i_{j+1}} + c_{i_{j+2}, i_r} b_{i_{j+2}}+ \cdots+c_{i_{r-1}, i_r} b_{i_{r-1}} + c_{i_j,\ i_r} $.
\end{center}
Hence $D \cdot V(\tau)=a_{i_j}+\sum_{l=j+1}^r a_{i_l}  b_{i_l} \geq s$ since $c_{i, j} \geq 0$ by Theorem \ref{12}.
$\hfill\square$\\

\noindent
We have the following applications of the above theorem.

\begin{cor}\label{2}
The divisor $D=\sum_{i=1}^{k}a_iD_{k+i}$ on $M_k$ is ample (respectively, nef) if and only if $a_i >  (\text{respectively, } \geq) \ 0$ for all $i=1, \ldots, k$. The anticanonical divisor $-K_{M_k}= D_1 + \ldots + D_{2k}$ is at most $2$-jet ample. Moreover, it is $2$-jet ample if and only if $c_{i, j}=0$ for all $i$ and $j$. Further, $M_k$ is Fano if and only if $\sum_{j=i+1}^k c_{i, j} \leq 1, i=1, \ldots, k-1$. Finally, the cotangent bundle on $M_k$ is never ample since $ K_{ M_k}$ is not ample.
\end{cor}

\begin{rmk}{\rm
Let $D=\sum_{i=1}^{k}a_iD_{k+i}$ be an ample $T$-invariant Cartier divisor on $M_k$. Then by the above ampleness criterion, $D-D_j$ is ample if and only if
\begin{itemize}
\item $a_j >1$ if $1 \leq j \leq k$ and
\item $a_{j-k}>1$ if $k+1 \leq j \leq 2k$.
\end{itemize}
For $j <j'$, $D-D_j-D_{j'}$ is not nef if and only if $j'=k+j$ and $a_j=1$. In fact, $D-D_j-D_{j'}$ is ample if and only if
\begin{itemize}
\item $a_j > 2$, when $j'=k+j$.
\item $b_j > 1$ and $b_{j'} > 1$, otherwise where
\[ b_l = \left\{ \begin{array}
{r@{\quad \quad}l}
a_l & l \leqslant k \\

 a_{l-k} & l > k
\end{array} \right. \]

\end{itemize}
This generalizes Mustata's criterion (see \cite{van}) in case of Bott tower.
}
\end{rmk}

\noindent
{\bf Generalization of ampleness criterion for toric vector bundles on Bott tower:}

Let us recall the notion of semi-stability of a nonzero torsion-free coherent sheaf $\mathcal{E}$ on $X$, where $X$ is a normal projective variety of dimension $n$ with an ample divisor $H$ on $X$. The slope of $\mathcal{E}$ is defined to be $\mu(\mathcal{E}):=\frac{\text{deg }(\mathcal{E})}{\text{rank }\mathcal{E}}$, where $\text{deg }(\mathcal{E})=c_1(\mathcal{E}) \cdot H^{n-1}$. The sheaf $\mathcal{E}$ is then said to be semi-stable if $\mu(\mathcal{F}) \leq \mu(\mathcal{E})$ for all proper nonzero subsheaves $\mathcal{F}\subsetneq \mathcal{E}$.

If a vector bundle $\mathcal{E}$ is ample, then it is known that determinant bundle $\text{det}(\mathcal{E})$ is ample,      the converse need not be true in general. But if $\mathcal{E}$ is semi-stable vector bundle on a smooth projective curve over algebraically closed field, the converse holds (see  \cite[Theorem $3.2.7$]{huybrechts2010geometry}). The following theorem generalizes this result for any nonsingular projective toric variety with some additional assumptions on $\mathcal{E}$.

\begin{prop}\label{7}
Let $\mathcal{E}$ be a toric bundle on a nonsingular projective toric variety $X=X(\Delta)$. Assume that $\mathcal{E}$ is semi-stable and discriminant $\Delta(\mathcal{E})=0$, where the discriminant of $\mathcal{E}$ is defined to be the characteristic class $\Delta(\mathcal{E}):=c_2(\mathcal{E}) - \frac{r-1}{2r} c_1(\mathcal{E})^2$. Then $\mathcal{E}$ is ample (respectively, nef) if and only if $\text{det}(\mathcal{E})$ is ample (respectively, nef).
\end{prop}

\noindent
{\bf Proof:} By \cite[Theorem $2.1$]{hering2010positivity}, $\mathcal{E}$ is ample (respectively, nef) if and only if $\mathcal{E}|_{V(\tau)}$ is ample (respectively, nef) for all walls $\tau$ (here wall means codimension $1$ cone). This is equivalent to say that $\text{deg}(\mathcal{E}|_{V(\tau)})> (\text{respectively, } \geq) 0$ for all walls $\tau$ by \cite[Theorem $3.2.7$]{huybrechts2010geometry} and \cite[Theorem $2.5$]{bruzzo2013restricting}. Now $\text{deg}(\mathcal{E}|_{V(\tau)})=\text{det}(\mathcal{E}) \cdot V(\tau)$. Hence we have  $\mathcal{E}$ is ample (respectively, nef) if and only if $\text{det}(\mathcal{E}) \cdot V(\tau) > (\text{respectively, }\geq) 0$  for all walls $\tau$ if and only if $\text{det}(\mathcal{E})$ is ample (respectively, nef) by toric Nakai criterion \cite[Theorem $2.18$]{Oda}. $\hfill\square$

\subsection{Criterion for nef and big line bundle on Bott tower:}
In this subsection we give a criterion for nef and big line bundle on Bott tower. A characterization for big divisors on an irreducible projective variety is given in \cite[Corollary $2.2.7$]{lazarsfeld2004positivity}. In case of toric varieties, we have the following:

\begin{prop}[Characterization of $T$-invariant big divisors]\label{3}
Let $X=X(\Delta)$ be a nonsingular projective toric variety. Let $D$ be a $T$-invariant Cartier divisor on X, then the following are equivalent:
\begin{itemize}
\item[(i)]$D$ is big.
\item[(ii)]For any $T$-invariant ample divisor $A$ on $X$, there exists a positive integer $n$ and $T$-invariant effective divisor $N$ on $X$, such that $nD \sim_{\text{lin}} A+N$.
\item[(iii)] same as in $(ii)$ for some $T$-invariant ample divisor $A$ on $X$.
\item[(iv)]There exists $T$-invariant ample divisor $A$ on $X$, a positive integer $n$  and $T$-invariant effective divisor $N$ on $X$, such that $nD \equiv_{\text{num}} A+N$.
\end{itemize}
\end{prop}

\noindent
{\bf Proof:} Assume $D$ is big. Since $A$ is $T$-invariant ample, there exists a positive integer $r$ such that both $rA$ and $(r+1)A$ are globally generated. So we can choose nonzero global sections $\chi^{m_1} \in H^0(X, \mathcal{O}_X(rA))$ and  $\chi^{m_2} \in H^0(X, \mathcal{O}_X((r+1)A))$. Then $rA \sim_{\text{lin}}(\chi^{m_1})_0$ and $rA \sim_{\text{lin}}(\chi^{m_2})_0$, where $H_r:=(\chi^{m_1})_0, \  H_{r+1}:=(\chi^{m_2})_0$ are effective divisors (\cite[Proposition II $7.7$]{hartshorne2013algebraic}). In fact, both the divisors $H_r$ and $H_{r+1}$ are also $T$-invariant. To see this, let $\{m_\sigma\}_{\sigma \in \Delta}$ be the Cartier data for $rA$. We have the isomorphism $\phi_\sigma: \mathcal{O}_X(rA)\mid_{U_\sigma} \simeq \mathcal{O}_{U_\sigma}$ given by multiplication by $\chi^{-m_\sigma}$. Then the Cartier data for $H_r$ is given by $\{m_1-m_\sigma\}_{\sigma \in \Delta}$, which shows that $H_r$ is $T$-invariant and similarly $H_{r+1}$ is also $T$-invariant.
Now by Kodaira's Lemma (\cite[Proposition $2.2.6$]{lazarsfeld14500positivity}) we have $H^0(X, \mathcal{O}_X(nD-H_{r+1})) \neq 0$ for some large positive integer $n$. Since $nD-H_{r+1}$ is $T$-invariant, arguing as above can choose nonzero global section $\chi^{m_3} \in H^0(X, \mathcal{O}_X(nD-H_{r+1}))$. Then $nD-H_{r+1} \sim_{\text{lin}} nD-H_{r+1}+\text{div}(\chi^{m_3} ) \geq 0$. Take $N'= nD-H_{r+1}+\text{div}(\chi^{m_3} )$ which is effective and $T$-invariant. Now  $nD-H_{r+1} \sim_{\text{lin}} N'$, which implies $nD \sim_{\text{lin}} H_{r+1}+N' \sim_{\text{lin}} (r+1)A +N' \sim_{\text{lin}} A +H_r+N'$. Taking $N=H_r+N'$, which is $T$-invariant effective we get  $nD \sim_{\text{lin}} A+N$. \\
\noindent
$(ii)\Rightarrow (iii)\Rightarrow (iv)$ Obvious.\\
\noindent
$(iv)\Rightarrow (i)$ Follows from the same argument as in (\cite[Corollary $2.2.7$]{lazarsfeld2004positivity}).
$\hfill\square$

\begin{cor}\label{4}
Let $X=X(\Delta)$ be a nonsingular projective toric variety. Let $D$ be a $T$-invariant Cartier divisor on $X$, then $D$ is nef and big if and only if there exists $T$-invariant effective divisor $N$ on $X$, such that $D-\frac{1}{n} N$ is ample for $n\gg 0$.
\end{cor}

\noindent
{\bf Proof:} Let $D$ be nef and big. Then by Proposition \ref{3} there exists a positive integer $k$, $T$-invariant ample Cartier divisor $A$ and $T$-invariant effective Cartier divisor $N$ on $X$, such that $kD \equiv_{\text{num}} A+N$. Take $n > k$, then $nD\equiv_{\text{num}} (n-k)D+A+N$. Since $D$ is nef and $A$ is ample, by \cite[Corollary $1.4.10$, Theorem $1.2.23$, Theorem $1.4.9$]{lazarsfeld14500positivity} $(n-k)D+A$ is ample. So $nD-N $ is ample, which implies $D-\frac{1}{n} N$ is ample for $n\gg 0$.

Converse direction follows by the same argument as in \cite[Example 2.2.19]{lazarsfeld14500positivity}.
$\hfill\square$

\begin{rmk}\label{19}\rm
Using the above criterion for bigness, for a $T$-invariant divisor $D=\sum_{i=1}^{k}a_iD_{k+i}$ on $M_k$ to be big one must have $a_k>0$. Hence the canonical divisor $K_{M_k}$ is never big.

\end{rmk}

\noindent
We now give a necessary and sufficient criterion for nef and big divisor on Bott tower.

\begin{thm}\label{5}
Let $D=\sum_{i=1}^{k}a_iD_{k+i}$ be a $T$-invariant divisor on $M_k$. Then $D$ is nef and big if and only if $a_k>0$, $a_i \geq 0$ for $1 \leq i < k$ and when  $a_i=0$ for some $i=1, \ldots, k-1$, then at least one of $\{c_{i, j} : j>i\}$ is nonzero.
\end{thm}

\noindent
{\bf Proof:} Let $D$ be nef and big. Then by Corollary \ref{2} and Remark \ref{19} we have $a_i \geq 0$ for $1 \leq i \leq k-1$ and $a_k>0$. Again Corollary \ref{4} implies that, there exists $T$-invariant effective divisor $N=\sum_{i=1}^{2k}b_iD_{k}$ on $M_k$ $(b_i \geq 0 \text{ for all } i)$, such that $D-\frac{1}{n} N$ is ample for $n\gg 0$. Write $N \sim_{\text{lin}} \sum_{i=1}^{k}c_iD_{k+i} $, where $c_k=b_k + b_{2k}$ and $c_i=b_i+b_{k+i}- \sum_{j=i+1}^kc_{i, j}b_j$, for $i=1, \ldots, k-1$. Take $d_i=a_i-\frac{1}{n} c_i$ and write $D-\frac{1}{n} N=\sum_{i=1}^{k}d_iD_{k+i}$. Now assume $a_i=0$ for some $i=1, \ldots, k-1$. By Corollary \ref{2}, $d_i > 0$ i.e., $(b_i+b_{k+i}- \sum_{j=i+1}^k c_{i, j}b_j) < 0$, which implies that at least one of $\{c_{i, j} : j>i\}$ must be nonzero.

Conversely, by Corollary \ref{2} we have $D$ is nef. Consider the $T$-invariant effective divisor $N=\sum_{i=1}^{2k}b_iD_i$ on $M_k$, where $\{ b_1 , b_2 , \ldots , b_k\}$ is a strictly increasing sequence of positive integers and $b_{k+i}=0$ for $i=1, \ldots, k$. Write $N \sim_{\text{lin}} \sum_{i=1}^{k}c_iD_{k+i} $, with $c_k=b_k $ and $c_i=b_i- \sum_{j=i+1}^kc_{i, j}b_j$, for $i=1, \ldots, k-1$. By Corollary \ref{4} it suffices to show that $D-\frac{1}{n} N=\sum_{i=1}^k(a_i- \frac{1}{n}c_i)$ is ample for $n\gg 0$. Since $a_k \geq 1$, for some positive integer $n_0$ we have $a_k- \frac{1}{n}c_k>0$ whenever $n > n_0$. If $a_i > 0$ for some $i \in \{1, \ldots, k-1\}$, then $ a_i - \frac{1}{n}c_i=a_i-\frac{1}{n}(b_i-\sum_{j=i+1}^k c_{i, j}b_j) > 0$ whenever $n > n_i$ for some large positive integer $n_i$. Now if $a_i =0$ for some $i \in \{1, \ldots, k-1\}$, by hypothesis $c_{i, j'} \neq 0$ for some $j'>i$. Then  $a_i - \frac{1}{n}c_i=- \frac{1}{n}(b_i- \sum_{j=i+1}^k c_{i, j}b_j )=- \frac{1}{n}(b_i-c_{i, j'}b_{j'})- \frac{1}{n} \left( \sum_{j=i+1,j \neq j'}^k c_{i, j}b_j \right)> 0$ since $b_i-c_{i, j'}b_{j'} < 0$ and $\sum_{j=i+1,j \neq j'}^k c_{i, j}b_j \geq 0$ by the choice of $b_i$'s. This competes the proof.
$\hfill\square$

Recall that a nonsingular projective algebraic variety is said to be weak Fano if its anticanonical divisor is nef and big. As an immediate corollary we obtain the following.

\begin{cor}\label{17}
$M_k$ is weak Fano if and only if $\sum_{j=i+1}^k c_{i, j} \leq 2, i=1, \ldots, k-1$.
\end{cor}

\begin{cor}
Every nef and big divisor on $M_k$ is ample if and only if $c_{i, j}=0$ for all $i$ and $j$.
\end{cor}
\noindent
{\bf Proof:} Let $c_{i, j} \neq 0$ for some $i, \ j$. Then consider $D= D_{k+1}+ \ldots + D_{k+i-1}+D_{k+i+1}+\ldots +D_{2k}$. Hence $D$ is nef and big by Theorem \ref{5}, but not ample by Corollary \ref{2}.

Converse is immediate from Theorem \ref{5} and Corollary \ref{2}.
$\hfill\square$

\subsection{Semi-stability of toric vector bundles on Hirzebruch surface:}

\noindent
Let $\mathcal{E}$ be a toric bundle on the Hirzebruch surface $M_2=\mathcal{H}_r$ with associated filtrations $\{E, E^{\rho}(i)\}_{\rho \in \Delta(1)}$. Then $c_1 ( \mathcal{E})= \sum_{i\in \mathbb Z, \rho \in \Delta(1)} i \text{ dim }E^{[\rho]}(i)D_\rho$ where $E^{[\rho]}(i)= E^{\rho}(i)/E^{\rho}(i+1)$ (see \cite[Remark $3.2.4$]{kly}). Fix a ample divisor $H=aD_3+bD_4$ $(a, b >0)$. Then $\text{deg }(\mathcal{E})= \text{deg }c_1 ( \mathcal{E})=c_1 ( \mathcal{E}) \cdot H= \sum_{i\in \mathbb Z} i \left( b \text{  dim }E^{[1]}(i) +a  \text{ dim }E^{[2]}(i)+b \text{ dim }E^{[3]}(i)+(a+br) \text{ dim }E^{[4]}(i) \right)$, since $D_1 \cdot D_3=D_3 \cdot D_3=D_2 \cdot D_4=0$, $D_2 \cdot D_3=D_4 \cdot D_3=D_1 \cdot D_4=1$ and $D_4 \cdot D_4=r$. Thus, $$\mu (\mathcal{E})= \frac{1}{\text{dim }E} \left( \sum_{i\in \mathbb Z} i \left( \text{ b dim }E^{[1]}(i) +a  \text{ dim }E^{[2]}(i)+b \text{ dim }E^{[3]}(i)+(a+br) \text{ dim }E^{[4]}(i)  \right) \right).$$

\begin{prop}\label{9} Let $\mathcal{E}$ be a toric bundle on $M_2=\mathcal{H}_r$. Then $\mathcal{E}$ is semi-stable with respect to  $H=aD_3+bD_4$ $(a, b >0)$ if and only if

$\frac{1}{\text{dim }F} \left( \sum_{i\in \mathbb Z} i \left( b \text{ dim }F^{[1]}(i) + a  \text{ dim }F^{[2]}(i)+b \text{ dim }F^{[3]}(i)+(a+br) \text{ dim }F^{[4]}(i) \right) \right) $ \\
$\leq \frac{1}{\text{dim }E} \left( \sum_{i\in \mathbb Z} i \left( \text{ b dim }E^{[1]}(i) +a  \text{ dim }E^{[2]}(i)+b \text{ dim }E^{[3]}(i)+(a+br) \text{ dim }E^{[4]}(i)  \right) \right)$\\
\noindent
for every proper subspace $F$ of $E$, where $F^{\rho}(i)=F \cap E^{\rho}(i)$ for any ray $\rho$.
\end{prop}

\noindent
{\bf Proof:} By \cite[Proposition $4.13$]{kool}, to check semi-stability of a toric vector bundle, it suffices to consider only toric subbundles of smaller rank which have been characterized in Proposition \ref{6}. Since for any nonsingular toric surface compatibility condition is automatically satisfied, this completes the proof.
$\hfill\square$

\noindent
\begin{cor}\label{14}
For $X=\mathcal{H}_r$, the tangent bundle $T_X$ is semi-stable with respect to $H=aD_3+bD_4$ if and only if $r=0, 1$. In particular, the semi-stability of $T_X$ does not depend on the choice of the polarization $H$.
 \end{cor}
\noindent
{\bf Proof:} We see that $$\mu (T_X)= \frac{\text{deg }(T_X)}{\text{rank }(T_X)}= \frac{2a+b(r+2)}{2}$$
Note that the only toric line subbundles of $T_X$ are $D_1, D_2, D_3, D_4$ and $D_2+D_4$ with $\mu=$ $b, \ a, \ b, \ a+br$ and $2a+br$ respectively. Then by Proposition \ref{9}, $T_X$ is semi-stable if and only if $br \leq 2(b-a)$ and $r \leq 2$ which happens if and only if $r$ is $0$ or $1$.
$\hfill\square$

Using Corollary \ref{14}, we give an example of a vector bundle satisfying the hypothesis of Proposition \ref{7}.

\begin{ex}{\rm
 Note that $T_X$ is semi-stable for $r=1$, $\Delta(T_X)=2$ and $\text{det}(T_X)=-K_X$ is ample but $T_X$ is not ample by Mori's theorem \cite[Remark $6.3.2$]{lazarsfeld14500positivity}.

Consider the line bundle $\mathcal{L}=\mathcal{O}_{X}(-D_1)$ on $X=\mathcal{H}_r$. Note that $c(T_X)=\prod_{i=1}^4(1+D_i)$ (see \cite[Example $3.2.3$]{kly}), so $c_1(T_X)=D_1+D_2+D_3+D_4$ and $c_2(T_X)=4$. Hence $\Delta(T_X))=c_2(T_X)-\frac{r-1}{2r} \ c_1^2(T_X)=2$ and we get that $\Delta(T_X \otimes \mathcal{L})=\Delta(T_X)+c_1(T_X) c_1(\mathcal{L})=0$. Note that $T_X \otimes \mathcal{L}$ is semi-stable if and only if $r=0$ or $1$. Let $\left( F, \{F^{\rho}(i) \}_{\rho \in \Delta(1)} \right)$ be the associated filtrations for the bundle $T_X \otimes \mathcal{L}$. For simplicity we write $F^j(i)$ to denote $F^{v_j}(i)$. Then $F^{1}(i)=E^{1}(i-1)$ and $F^{j}(i)=E^{j}(i)$ for $j =2, 3, 4$.
\begin{center}
$
\text{dim}(F^{[1]}(i)) = \left\{ \begin{array}{ccc}

1 & i = 1\\
1 & i=2 \\
 0 & \text{otherwise}
\end{array} \right.
$
and
$
\text{dim}(F^{[j]}(i)) = \left\{ \begin{array}{ccc}

1 & i = 0\\
1 & i=1 \\
 0 & \text{otherwise}
\end{array} \right.
$
for $j =2, 3, 4$.
\end{center}

\noindent
$\text{det}(T_X \otimes \mathcal{L})=\sum_{i \in \mathbb Z, \ j=1}^4 i \text{ dim}(F^{[j]}(i)) D_j=3D_1+D_2+D_3+D_4$. Now $\text{det}(T_X \otimes \mathcal{L})$ is ample if and only if $r<4$ by Corollary \ref{2}. Hence $T_X \otimes \mathcal{L}$ is ample by Proposition \ref{7}.
}
\end{ex}

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Revision as of 09:59, 6 April 2016 by Robert (talk | contribs) (Robert moved page Fort Mitchell to Fort Mitchel)
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Satellite Image of Fort Mitchell from Google Earth
Fort Mitchell is the present name for Fort Westmoreland. It was renamed in 1938 when Ireland regained the Treaty Ports from the British. It is no longer in use by the Irish Defence Forces. It is located on Spike Island in Cork Harbour.
Fort Mitchell was one of four forts constructed to protect Cork Harbour. Construction of the fort began in 1790. By virtue of Cork Harbour being a Treaty Port, it remained occupied by British Forces until July 1938 when control was handed over to the Irish Defence Forces.
Fort Westmoreland was named after the then Lord Lieutenant, the Earl of Westmoreland. When handed over to the Irish Defence Forces it was renamed Fort Mitchell after John Mitchell who was an Irish nationalist activist and political journalist interred on the island prior to deportation to Van Diemen's Land in 1848.
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This solid University of South Florida diploma frame features Mahogany Braid hardwood moulding and comes double matted, black on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 17 1/4" x 19") Master's degrees earned summer of 2009 and after and all PhD Degrees: 11" x 14". (Approximate overall frame size: 20" x 23") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 18 1/2" x 20 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-MBr-S-BG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Cherry Reverse hardwood moulding and comes double matted, green on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-CR-S-GrG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Cherry Lacquer hardwood moulding and comes double matted, green on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-CL-S-GrG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Rosewood hardwood moulding and comes double matted, green on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-R-S-GrG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Mahogany Lacquer hardwood moulding and comes double matted, green on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-ML-S-GrG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Rosewood w/Gold Lip hardwood moulding and comes double matted, green on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-RGL-S-GrG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Mahogany Braid hardwood moulding and comes double matted, green on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 17 1/4" x 19") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 20" x 23") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 18 1/2" x 20 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-MBr-S-GrG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Cherry Reverse hardwood moulding and comes double matted, black suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-CR-S-BSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Cherry Lacquer hardwood moulding and comes double matted, black suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-CL-S-BSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Rosewood hardwood moulding and comes double matted, black suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-R-S-BSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Mahogany Lacquer hardwood moulding and comes double matted, black suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-ML-S-BSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Rosewood w/Gold Lip hardwood moulding and comes double matted, black suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-RGL-S-BSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Mahogany Braid hardwood moulding and comes double matted, black suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 17 1/4" x 19") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 20" x 23") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 18 1/2" x 20 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-MBr-S-BSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Cherry Reverse hardwood moulding and comes double matted, green suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-CR-S-GrSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Cherry Lacquer hardwood moulding and comes double matted, green suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-CL-S-GrSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Rosewood hardwood moulding and comes double matted, green suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-R-S-GrSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Mahogany Lacquer hardwood moulding and comes double matted, green suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-ML-S-GrSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Rosewood w/Gold Lip hardwood moulding and comes double matted, green suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-RGL-S-GrSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Mahogany Braid hardwood moulding and comes double matted, green suede on gold. The official USF seal and name are elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 17 1/4" x 19") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 20" x 23") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 18 1/2" x 20 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-MBr-S-GrSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Cherry Reverse hardwood moulding and comes double matted, black suede on gold. The 24K Plated-Medallion features the official USF seal, and the school name is elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-CR-S-BSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Rosewood hardwood moulding and comes double matted, black suede on gold. The 24K Plated-Medallion features the official USF seal, and the school name is elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-R-S-BSueG) See all of the USF diploma frame options here.
This solid University of South Florida diploma frame features Mahogany Lacquer hardwood moulding and comes double matted, black suede on gold. The 24K Plated-Medallion features the official USF seal, and the school name is elegantly embossed in gold leaf directly onto our premium archival matting. A pane of UV protective glass finishes it all off and keeps your USF diploma looking as beautiful as the day it was earned. Comes with a 100% Lifetime Guarantee, making it the perfect, worry-free graduation gift. [USF Bachelors degrees: 8.5" x 11" - horizontal. (Approximate overall frame size: 16 1/4" x 18") Master's degrees earned summer of 2009 and after, and all PhD Degrees: 11" x 14". (Approximate overall frame size: 19" x 22") Master's degrees earned spring of 2009 and before: 10" x 13". (Approximate overall frame size: 17 1/2" x 19 1/2") Specify degree accordingly in the drop-down menu above] Learn more about why our archival materials and UV glass are crucial when framing a diploma. (Product# USF-ML-S-BSueG) See all of the USF diploma frame options here.