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\section{Introduction}
\label{sec:intro}

\subsection{History and Motivation}
Constructing global sections of line bundles or having effective control over such is an ancient and consistently difficult problem in algebraic geometry. In the last decades, much of the effort directed towards this problem was guided by the conjectures of Fujita (for global generation and very ampleness  \cite{fujita_polarized_1987}) and Mukai (for the study of higher syzygies, see \cite[Conjecture~4.2]{ein_syzygies_1993}).

The main purpose of our work is to study effective global generation of convex integral linear combinations of ample divisors while being able to determine precise Fujita-type bounds in a number of cases. In particular, we succeed in constructing examples with the following prescribed parameters: dimension, fundamental group, and global generation behavior. We essentially  exhaust the range of these invariants simultaneously given the global generation behavior predicted by Fujita's freeness conjecture.

Fujita's conjectures on global generation anticipate that given a smooth projective variety $X$ equipped with an ample Cartier divisor $L$, the adjoint divisor $K_X+mL$ should be base point free provided $m\geq \dim(X)+1$.  This  is a classical consequence of Riemann--Roch if $\dim(X)=1$, and has been demonstrated for $\dim(X)\leq 5$ \cites{reider_vector_1988,ein_global_1993,kawamata_fujitas_1997,helmke_fujitas_1997,ye_fujitas_2020}. For arbitrary dimension, there exist non-trivial global generation results due to Angehrn--Siu \cite{angehrn_effective_1995}, Heier \cite{heier_effective_2002},  and Ghidelli and Lacini \cite{ghidelli_logarithmic_2021}. It is important to remember that these bounds are uniform but nevertheless not linear in $\dim(X)$.

Effective global generation behavior of divisors on a given variety has not been explicitly known except in a handful of cases. Fujita's freeness conjecture is sharp for hyperplane divisors on projective spaces $\mathbb{P}^n$; on the other end of the spectrum, a classical result of Lefschetz shows that given an ample divisor  $L$ on an abelian variety $X$, the divisor $2L$ is globally generated independently of the dimension of $X$.
Related results have been obtained by Pareschi--Popa \cite[Theorem 5.1]{pareschi_regularity_2003} on the global generation of $2K_X + 2L$ for a nef and big divisor $L$ on an irregular variety $X$ with finite Albanese morphism.  In the context of varieties $S$ with numerically trivial canonical divisors, there are results for the Hilbert scheme $\Hilb^2(S)$ of subschemes of length $2$, by Riess~\cite{ries_non-divisorial_2021} when $S$ is a K3 surface of degree $2$,
and by K\"uronya--Mustopa \cite{kuronya_effective_2022} when $S$ is an abelian surface.

In order to be able to talk about intermediate bounds on effective global generation and treat global generation in higher codimensions, the authors of \cite{kuronya_effective_2022} introduced Fujita numbers  for coherent sheaves with a view towards the codimension of the base locus.
It is this line of thought that we follow, except that we focus solely on the Fujita number
describing global generation itself --- being the closest in spirit to the freeness conjecture --- while introducing the (topological) fundamental group of the variety  as an extra parameter, and concentrating on the finer variant using convex linear combinations instead of multiples of a given ample divisor.

\subsection{Convex Fujita numbers}
As mentioned above, we will measure effective positivity of  line bundles via the global generation of the associated sequence of adjoint ones, as  suggested by Fujita's conjectures.  Throughout this paper we will work with varieties over ${\mathbb C}$.

\begin{defi}
The \textbf{convex Fujita number} of a smooth projective variety $X$ is the minimal $m \geq 0$ such that  for all $s \geq m$ and any ample divisors $L_1, \ldots, L_s$ on $X$ the adjoint divisor
\[
K_X + L_1 + \ldots + L_s
\]
is globally generated. We will denote the convex Fujita number by $\conFN(X)$, or,
if no such $m$ exists, then we set $\conFN(X) = \infty$ (but this does not occur, see \pref{prop:FNisFinite}).
\end{defi}

Fujita conjectured in \cite{fujita_polarized_1987} that for a smooth projective variety $X$ of dimension $n$ and any ample divisor $L$ on $X$ the multiple adjoint divisor $m( K_X + t L)$ is globally generated if $m + t > n+1$ and $K_X+ tL$ is nef. Fujita moreover shows in the same paper that $K_X + tL$ is nef for $t \geq n+1$. As a result, only the case $m=1$ has been traditionally referred to as  follows.

\begin{conj}[\textbf{Fujita's freeness conjecture} \cite{fujita_polarized_1987}]
\label{conj:FujitaFreeness}
Let $X$ be a smooth projective variety and let $L$ be an ample divisor on $X$. Then for all $m \geq \dim(X) + 1$ the adjoint divisor $K_X + m L$ is globally generated.
\end{conj}

Fujita's freeness conjecture follows directly from the Riemann--Roch theorem for curves; for surfaces it is a quick consequence of Reider's theorem obtained by vector bundle techniques  \cite{reider_vector_1988},  Ein and Lazarsfeld in \cite{ein_global_1993} showed it for threefolds, Kawamata in \cite{kawamata_fujitas_1997} for $4$-folds and Ye and Zhu in \cite{ye_fujitas_2020} for 5-folds. All results for $\dim(X) \geq 3$ rely on vanishing theorems and non-klt center methods. The conjecture is currently open in dimensions six and above.

Uniform bounds for $m$ have been proven such that $K_X + m L$ is globally generated for all ample divisors $L$ on all smooth projective varieties $X$ of a given dimension $n$. Angehrn and Siu show in \cite[Theorem 0.1]{angehrn_effective_1995}  that $m \geq (n^2 + n + 2)/2$ suffices, while Heier \cite[Theorem 3.1]{heier_effective_2002} improved the bound to $m \geq (e + 1/2)n^{4/3} + 1/2 n^{2/3} + 1$, where $e = \exp(1)$ is Euler's number. More recently, Ghidelli and Lacini proved in  \cite[Theorem 1.1]{ghidelli_logarithmic_2021} an asymptotically much better bound of
\[
m \geq \max \{n + 1, n(\log \log(n) + 2.34)\}.
\]
We will discuss  in \secref{sec:FujitaFinite} the impact on the convex Fujita number of some of the methods and results  on Fujita's freeness conjecture as recalled above.

A numerical version of \conjref{conj:FujitaFreeness} was proposed by Helmke \cite[Conjecture 1.2]{helmke_fujitas_1997} as follows.
\begin{conj}[\textbf{numerical Fujita's freeness conjecture} \cite{helmke_fujitas_1997}]
\label{conj:numericalFujitaFreeness}
Let $X$ be a smooth projective variety of dimension $\dim(X) = n$ and let $L$ be an ample divisor on $X$ such that
\begin{enumerate}[label=(\roman*),align=left,labelindent=0pt,leftmargin=*,widest = (iii)]
\item
$(L^n) > n^n$, and
\item
for all irreducible cycles $Z \subseteq X$ of dimension $\dim(Z) = d<n$ we have $(L^d \cdot  Z) \geq n^d $.
\end{enumerate}
Then the adjoint divisor $K_X+ L$ is globally generated.
\end{conj}

The numerical Fujita's freeness conjecture implies $\conFN(X) \leq \dim(X) + 1$ for all $X$ (see
\pref{prop:FNisFinite} for the elementary argument showing a weaker but unconditional bound), and this bound in turn obviously implies Fujita's freeness conjecture. We may wonder whether
\[
\conFN(X) \leq \dim(X) + 1
\]
always holds.  As we will see, it holds for all the examples constructed in this paper whose convex Fujita number is calculated precisely.

\subsection{Varieties with prescribed convex Fujita numbers}

\begin{defi}[Fujita simple and Fujita extreme varieties]
We say that a smooth projective  variety $X$ is
\begin{itemize}
	\item \emph{Fujita simple} if  $\conFN(X) = 0$, and
	\item \emph{Fujita extreme} provided $\conFN(X) \geq \dim(X) + 1$.
\end{itemize}

\end{defi}

\begin{ex}
For $X = \mathbb{P}^n$ all ample line bundles are isomorphic to ${\mathcal O}(a)$ for some $a> 0$.  It follows at once that $\conFN(\mathbb{P}^n) = n+1$, so Fujita extreme varieties exist in all dimensions.
\end{ex}

\begin{ex}
Riemann-Roch implies that all smooth projective curves $C$ have $\conFN(C) = 2$ regardless of the genus of $C$. The line bundles $\mathscr{L}$ on $C$ such that $\omega_C \otimes \mathscr{L}$ is not globally generated are precisely the line bundles $\mathscr{L} = {\mathcal O}_C(P)$ for an arbitrary point $P \in C$.
\end{ex}

\begin{rem}
The canonical bundle of a Fujita simple variety is by definition globally generated and thus nef. In particular, Fujita simple varieties are minimal.
\end{rem}

\begin{ex}
Its a classical theorem of Lefschetz that on an  abelian variety $A$ already $\mathscr{L}^{\otimes 2}$ is globally generated for any ample line bundle $\mathscr{L}$. The proof of Lefschetz's theorem generalizes to estimate the convex Fujita number of abelian varieties by $\conFN(A) \leq 2$  This was proved in \cite[Theorem 1.1]{bauer_tensor_1996}. It is
based on an application of the theorem of the square
\[
\mathscr{L}^{\otimes 2} \simeq t_{x}^\ast \mathscr{L} \otimes t_{-x}^\ast \mathscr{L},
\]
Kodaira vanishing and Riemann-Roch $\rh^0(A,\mathscr{L}) = \chi(A,\mathscr{L}) = \frac{1}{g!} (\mathscr{L}^g) > 0$
(applied to the translates $t_{x}^\ast \mathscr{L}$ and $t_{-x}^\ast \mathscr{L}$ instead of $\mathscr{L}$), and the product map
\[
\bigoplus_{x \in A} \rH^0(A,t_{x}^\ast \mathscr{L} ) \otimes \rH^0(A,t_{-x}^\ast \mathscr{L}) \to  \rH^0(A,\mathscr{L}^{\otimes 2}).
\]
\end{ex}

We recall that a group $\pi$ is called \textbf{projective} if it is isomorphic to the (topological) fundamental group $\pi_1(X)$ of a smooth projective variety $X$.

\begin{thmABC}[see \pref{prop:FNsimpleSurface} and \pref{prop:FNsimpleVariety}]
\label{thmABC:FNsimple}
Let $\pi$ be a projective group, and let $n \geq 2$ be an integer. There is a Fujita simple smooth connected projective variety $X$ of dimension $n$ with
\begin{enumerate}[label=(\roman*),align=left,labelindent=0pt,leftmargin=*,widest = (iii)]
\item
$X$ is of general type, and
\item
$\pi_1(X)$ is isomorphic to $\pi$.
\end{enumerate}
\end{thmABC}
The varieties $X$ that prove \tref{thmABC:FNsimple} are constructed in two ways. The first method starts with the surface case $n=2$ (dealt with in \pref{prop:FNsimpleSurface}) and then takes the product of a Fujita simple surface and a simply connected Fujita simple variety, more precisely a hypersurface in  $\mathbb{P}^{n-1}$ of degree at least $n+1$.  This covers dimensions $n \geq 4$.

The second method covers $n \geq 3$ and constructs $X$ as  a cyclic cover $f: X \to Y$ that totally ramifies along a smooth ample divisor, such that the degree $d = \deg(f)$ is sufficiently large. The variety $Y$ is a complete intersection of the correct dimension $n$ in $S \times \mathbb{P}^n$ with a smooth projective variety $S$ that has the correct fundamental group $\pi_1(S) = \pi$. Such an $S$ exists because $\pi$ is assumed to be a projective group.

More precisely, analyzing complete intersections in $\mathbb{P}^N$ leads to examples for $n \geq 3$ and for $(n, k) = (2,0)$ of the following theorem. The remaining examples in dimension $n=2$ are obtained as the blow-up of $\mathbb{P}^2$ in $3-k$ points.

\begin{thmABC}[see \tref{thm:FNsimplyconnected}]
\label{thmABC:FNsimplyconnected}
Let  $n \geq 2$ be an integer, and let $0 \leq k \leq n+1$. There is a simply connected smooth connected projective variety $X$ of dimension $n$ with convex Fujita number $\conFN(X) = k$. Moreover, if $k=0$, then $X$ can be chosen to be of general type.
\end{thmABC}

The behavior of convex Fujita numbers in products is mysterious in general, but easy in the following special case. It will be applied in the case of a simply connected variety $Y$.

\begin{propABC}[see \cref{cor:FNproductwithPn}]
\label{propABC:FNproductwithPn}
Let $X$ be a smooth projective variety, and let $Y$ be a smooth projective variety with $\rH^{1}(Y,{\mathcal O}_{Y}) = 0$.  Then
\[
\conFN(X \times Y) = \max\{\conFN(X), \conFN(Y)\}.
\]
\end{propABC}

The K\"unneth formula for $\pi_1$, \pref{propABC:FNproductwithPn} and the case $k=0$ of \tref{thmABC:FNsimplyconnected} reduce \tref{thmABC:FNsimple} to the case of dimension $n= 2$ and $3$. Even more follows by combining of \tref{thmABC:FNsimple} with the full version of \tref{thmABC:FNsimplyconnected}.

 Our next result indicates that the topological invariant $\pi_1(X)$ alone is not sufficient to control positivity properties of adjoint line bundles.

\begin{thmABC}[see \tref{thm:anyFNanyPi1}]
\label{thmABC:anyFNanyPi1}
Let $\pi$ be a projective group, and let $n \geq 2$ be an integer, and let $0 \leq k \leq n-1$. There is a smooth connected projective variety $X$ of dimension $n$ with the following properties:
\begin{enumerate}[label=(\roman*),align=left,labelindent=0pt,leftmargin=*,widest = (iii)]
\item
$\pi_1(X)$ is isomorphic to $\pi$, and
\item
$\conFN(X) = k$.
\end{enumerate}
\end{thmABC}

The surface with prescribed fundamental group and convex Fujita number $1$ is obtained as the blow up in one point of a carefully chosen surface with the same fundamental group and Fujita number $0$. The threefold with convex Fujita number $\conFN(X) = 2$ arises as $X = S \times \mathbb{P}^1$ with a surface $S$ with the same fundamental group and convex Fujita number $\conFN(S) \leq 2$. The threefold $X'$ with convex Fujita number $\conFN(X') = 1$ arises as a branched double of the threefold $X$ with  $\conFN(X) \leq 2$.

\smallskip

\tref{thmABC:anyFNanyPi1} has the following obvious corollary.

\begin{corABC}
	The precise value of the Fujita number $\conFN(X)$ does not depend on the fundamental group $\pi_1(X)$ alone.
\end{corABC}

\subsection{Acknowledgements}
The authors acknowledge support by Deutsche  Forschungsgemeinschaft  (DFG) through the Collaborative Research Centre TRR 326 "Geometry and Arithmetic of Uniformized Structures", project number 444845124.  Part of this work was done while the third author attended the workshop "Birational Complexity of Algebraic Varieties" at the Simons Center for Geometry and Physics, and he would like to thank the organizers and staff for the hospitality and stimulating atmosphere.

\section{Preliminaries on convex Fujita numbers}
\label{sec:FNprelims}

\subsection{Finiteness}
\label{sec:FujitaFinite}

We first show that convex Fujita numbers are finite. We start with surfaces, $3$-folds and toric varieties in order to review that the typical techniques in the Fujita freeness conjecture specific to these dimensions also bound the convex Fujita number.

\begin{prop}
\label{prop:FNsurfacesReider}
Let $X$ be a smooth projective surface.
\begin{enumerate}[align=left,labelindent=0pt,leftmargin=*]
\item
\label{propitem:reider1}
The convex Fujita number of $X$ is bounded by $\conFN(X) \leq 3$.
\item
\label{propitem:reider2}
If $\conFN(X) = 3$, then there exist an ample divisor $L$ on $X$ with $(L^2) = 1$.
\item
\label{propitem:reider3}
If the intersection pairing on the N\'eron-Severi lattice $\NS(X)$ is even, then $\conFN(X) \leq 2$.
\item
\label{propitem:reider4}
If the canonical divisor is numerically equal to $2 D$
with $D \in \Div(X)$, then $\conFN(X) \leq 2$.
\end{enumerate}
\end{prop}
\begin{proof}
By the Riemann-Roch formula, the assumption in \eqref{propitem:reider4} implies the assumption in \eqref{propitem:reider3} and clearly thus both then follow from \eqref{propitem:reider1} and \eqref{propitem:reider2}.

Let us prove \eqref{propitem:reider1}. We have to show a bound on $m$ such that for ample divisor $L_1, \ldots, L_m$ the adjoint divisor $K_X + L$ with $L = L_1 + \ldots + L_m$ is globally generated. If $m \geq 3$, then $(L^2) \geq 9$ and by \cite[Theorem 1 (i)]{reider_vector_1988} base points can only occur if there is an effective divisor $C$ such that $(C^2) \leq 0$ and $(L \cdot C) - (C^2) = 1$. But $(L \cdot C) \geq m$ since the $L_i$ are ample, contradiction.

For \eqref{propitem:reider2} we must have a base point of $K_X + L$ when $m=2$. Reider's method as in the proof of \eqref{propitem:reider1} still works unless $(L^2) \leq 4$, which implies $(L_i \cdot L_j) = 1$ for all $i, j$. This shows \eqref{propitem:reider2}.
\end{proof}

The following consequence of Reider's method will be used in our proof of \tref{thmABC:FNsimple}.

\begin{prop}
\label{prop:FNsurfacesReider divisible intersection numbers}
Let $X$ be a smooth projective surface.
If the intersection pairing on the N\'eron-Severi lattice $\NS(X)$  takes values in $d{\mathbb Z}$ for some $d \geq 5$, then the convex Fujita number of $X$ is bounded by $\conFN(X) \leq 1$.
\end{prop}
\begin{proof}
We must show that for any ample divisor $L$ on $X$ the adjoint divisor $K_X + L$ is globally generated. Since by assumption $(L^2) \geq d \geq 5$, we can apply \cite[Theorem 1 (i)]{reider_vector_1988} and must exclude the exceptional case: there is an effective divisor $C$ such that $(C^2) \leq 0$ and $(L \cdot  C) - (C^2) = 1$. But by assumption $d$ divides $(L \cdot  C) - (C^2)$, so we are done.
\end{proof}

\begin{prop}
Let $X$ be a smooth projective threefold. Then the convex Fujita number is bounded by $\conFN(X) \leq 4$.
\end{prop}
\begin{proof}
Let $\mathscr{L}$ be a sum $L_1 + \ldots + L_m$ of $m \geq 4$ ample divisors $L_i$ on $X$. Then for all $i, j, k$ and all surfaces $S$ and all curves $C$ in $X$ we have
\[
(L_i \cdot L_j \cdot L_k) \geq 1, \quad (L_i \cdot L_j \cdot S) \geq 1, \quad (L_i \cdot C) \geq 1,
\]
so that
\[
(L^3) \geq m^3 > 27, \quad (L^2 \cdot S) \geq m^2\geq 9, \quad (L \cdot C) \geq m \geq 3.
\]
The estimate $\conFN(X) \leq 4$ now follows from \cite[Theorem 5.2]{helmke_fujitas_1997}.
\end{proof}

Fujita's freeness conjecture has been proved for toric varieties in  \cite[Theorem 0.3]{laterveer_linear_1996}. The bound for the convex Fujita number of toric varieties has also been established.

\begin{prop}
For a toric variety $X$ the convex Fujita number is bounded by
\[
\conFN(X) \leq \dim(X) + 1.
\]
Equality holds only for $X = \mathbb{P}^n$.
\end{prop}
\begin{proof}
In view of $\omega_X = {\mathcal O}_X(-\sum_i D_i)$ with $D_i$ being the torus invariant prime divisors, the result follows at once from \cite[Theorem 0.3]{mustata_vanishing_2002}\footnote{Musta\c{t}\u{a} describes this theorem as a strong version of Fujita's freeness conjecture. The notion of convex Fujita number provides a conceptual framework for this kind of strengthening of Fujita's conjecture. The result can also be extracted from \cite{laterveer_linear_1996}, but the chosen reference is more direct.}.
\end{proof}

An explicit finiteness statement for the convex Fujita number follows from work of Angehrn and Siu. It is noteworthy that the bound is uniform in the dimension.

\begin{prop}
\label{prop:FNisFinite}
For a smooth projective variety of dimension $n$ the convex Fujita number is bounded as
\[
\conFN(X) \leq  \frac{1}{2}(n^2 + n + 2).
\]
\end{prop}
\begin{proof}
Let $L = L_1 + \ldots + L_m$ be a sum of $m \geq \frac{1}{2}(n^2 + n + 2)$ ample  divisors on $X$. For any irreducible cycle $W$ on $X$ of dimension $d$ and any map $\sigma : \{1,\ldots, d\} \to \{1, \ldots, m\}$ we thus find
\[
(L_{\sigma(1)} \cdot  \ldots \cdot  L_{\sigma(d)} \cdot  W) \geq 1,
\]
so that $(L^d \cdot W) \geq m^d$. It follows that the criterion of \cite[Theorem 0.1]{angehrn_effective_1995} for global generation of $K_X + L$ is satisfied.
\end{proof}

\subsection{Fujita numbers of products}
Let $X$ and $Y$ be smooth projective varieties. For ample line bundles $\mathscr{L}$ on $X$ and $\mathscr{M}$ on $Y$ the line bundle $\mathscr{L} \boxtimes \mathscr{M}$ on $X \times Y$ is ample. Let $\mathscr{L}_i$ (resp.\ $\mathscr{M}_i$) be ample line bundles on $X$ (resp.\ on  $Y$) for $i = 1, \ldots, m$.  If the line bundle
\begin{equation}
\label{eq:kuenneth}
\omega_{X \times Y} \otimes \bigotimes_{i=1}^m (\mathscr{L}_i \boxtimes \mathscr{M}_i) = (\omega_X \otimes \bigotimes_{i=1}^m \mathscr{L}_i ) \boxtimes (\omega_Y \otimes \bigotimes_{i=1}^m \mathscr{M}_i)
\end{equation}
is globally generated, then also its restriction $\omega_X \otimes \bigotimes_{i=1}^m \mathscr{L}_i$  (resp.\ $\omega_Y \otimes \bigotimes_{i=1}^m \mathscr{M}_i$) to the fiber of the projection $X \times Y \to X$ (resp.\ $X \times Y \to Y$) is globally generated. This immediately shows the following lemma.

\begin{lem}
\label{lem:FNforProduct}
Let $X$ and $Y$ be smooth projective varieties. Then
\[
\conFN(X \times Y) \geq \max\{\conFN(X), \conFN(Y) \}.
\]
\end{lem}

The estimate can be improved to an equality in some favorable cases.

\begin{prop}
\label{prop:FNforProduct}
Let $X$ and $Y$ be smooth projective varieties such that the abelian varieties $\Pic^0_X$ and $\Pic^0_Y$ have no common nontrivial isogeny factor. Then  the following holds.
\begin{enumerate}[align=left,labelindent=0pt,leftmargin=*]
\item
\label{propitem:FNforProduct1}
$\Pic(X \times Y) = \Pic(X) \times \Pic(Y)$, and
\item
\label{propitem:FNforProduct2}
$\conFN(X \times Y) = \max\{\conFN(X), \conFN(Y) \}.$
\end{enumerate}
\end{prop}
\begin{proof}
We first show that \eqref{propitem:FNforProduct1} implies \eqref{propitem:FNforProduct2}. The product decomposition for $\Pic(-)$ is natural by restriction to fibers, hence also the nef cones and its interior, the ample cone, are products of the respective cones of the factors. Now arguing as in the proof of \lref{lem:FNforProduct}, it remains to see that $\omega_{X \times Y} \otimes \bigotimes_{i=1}^m (\mathscr{L}_i \boxtimes \mathscr{M}_i)$ is globally generated if the restrictions to the fibers of both projections are globally generated. This follows at once by the K\"unneth formula applied to \eqref{eq:kuenneth}.

For \eqref{propitem:FNforProduct1} we analyze the Leray spectral sequence for $\mathbb{G}_m$ along the projection $\text{pr}: X \times Y \to X$. The low degree terms yield an exact sequence
\[
0 \to \Pic(X) \xrightarrow{\text{pr}^\ast} \Pic(X \times Y) \to \rH^0(X,\mathbb{R}^1\text{pr}_\ast \mathbb{G}_m) \xrightarrow{d_2^{0,1}} \rH^2(X,\mathbb{G}_m)  \xrightarrow{\text{pr}^\ast} \rH^2(X \times Y, \mathbb{G}_m).
\]
Since the choice of a point $y \in Y$ and the map $i(x) = (x, y)$ splits the projection, the map $\text{pr}^*$ admits a retraction. Hence the boundary map $d_2^{0,1}$ is the zero map and we have a short exact sequence
\[
0 \to \Pic(X) \xrightarrow{\text{pr}^\ast} \Pic(X \times Y) \to \rH^0(X,\mathbb{R}^1\text{pr}_\ast \mathbb{G}_m)  \to 0.
\]
The sheaf $\mathbb{R}^1\text{pr}_\ast \mathbb{G}_m$ is represented by the Picard variety of $Y$, hence
\[
\rH^0(X,\mathbb{R}^1\text{pr}_\ast \mathbb{G}_m) = \Hom(X,\Pic_Y).
\]
Now we fix a point $x \in X$. Evaluation in $x$ and constant maps, as well as the Albanese property describe a canonical splitting
\[
\Hom(X,\Pic_Y) = \Pic(Y) \times \Hom((X, x), (\Pic_X,0)) = \Pic(Y) \times \Hom(\Alb_X, \Pic^0_Y).
\]
Because Albanese and $\Pic^0$ are dual abelian varieties and thus share the same isogeny factors, our assumption is precisely that the second factor vanishes. This proves \eqref{propitem:FNforProduct1}.
\end{proof}

We now prove \pref{propABC:FNproductwithPn} from the introduction.

\begin{cor}
\label{cor:FNproductwithPn}
Let $X$ be a smooth projective variety, and let $Y$ be a smooth projective variety with $\rH^{1}(Y,{\mathcal O}_{Y}) = 0$.
Then
\[
\conFN(X \times Y) = \max\{\conFN(X), \conFN(Y)\}.
\]
\end{cor}
\begin{proof}
Since $\rH^{1}(Y,{\mathcal O}_{Y}) = 0$, we have that $\Pic_Y^0$ is trivial, so \pref{prop:FNforProduct} proves the claim.
\end{proof}

\section{Simply connected varieties}
\label{sec:FujitaSimplyConnected}

\subsection{Complete intersections} In this section let $X$ be a smooth complete intersection in $\mathbb{P}^{n+r}$ of dimension $n\geq 2$ of hypersurfaces  of degree $d_i$ for $i=1, \ldots, r$.

\begin{prop}
\label{prop:FNcompleteintersection}
If $n \geq 3$, then the complete intersection $X \hookrightarrow \mathbb{P}^{n+r}$ of multidegree $(d_1, \ldots, d_r)$ has convex Fujita number
\[
\conFN(X) = \max\big\{0,  (n+r+1) - \sum_{i=1}^r  d_i \big\}.
\]
\end{prop}
\begin{proof}
By the Lefschetz hyperplane theorem for $\Pic(-)$,
see \cite[Exp.~XII Cor.~3.6]{grothendieck_cohomologie_1965},
the group $\Pic(X)$ is generated by ${\mathcal O}(1)|_X$. The line bundle $\mathscr{L} = {\mathcal O}(a)|_X$ is ample if and only if $a \geq 1$, and it is globally generated if and only if $a \geq 0$.

The adjoint bundle $\omega_X \otimes \mathscr{L}$ for a product  $\mathscr{L} = \mathscr{L}_1 \otimes \ldots \otimes \mathscr{L}_m$ of ample line bundles $\mathscr{L}_i = {\mathcal O}(a_i)|_X$ is by adjunction
\[
\omega_X \otimes \mathscr{L} = {\mathcal O}\big(- (n+r+1) + \sum_{i=1}^m a_i + \sum_{i=1}^r d_i \big)
\]
For a fixed $m$ all possible $a_i \geq 1$ lead to globally generated adjoint bundles if and only if the special case $a_i = 1$ for all $i=1, \ldots, m$ leads to a globally generated line bundle, equivalently if
\[
m  \geq (n+r+1) -  \sum_{i=1}^r d_i.
\]
This translates into the claimed formula for $\conFN(X)$.
\end{proof}

\subsection{Simply connected surfaces} While complete intersections can treat dimension $\geq 3$, for surfaces we argue with explicit examples.

\begin{prop}
\label{prop:FNsurfaces simply connected}
Let $0 \leq k \leq 3$ be an integer. Then there is a simply connected smooth projective surface $X$  with convex Fujita number $k$.
\begin{enumerate}[align=left,labelindent=0pt,leftmargin=*]
\item
\label{propitem:FNsurface sc1}
More concretely, for $1 \leq k \leq 3$ the blow up of $\mathbb{P}^2$  in $3-k$ points has $\conFN(X) = k$.
\item
\label{propitem:FNsurface sc2}
A very general hypersurface $X$ in $\mathbb{P}^3$ of degree $d \geq 5$ is simply connected, has convex Fujita number $\conFN(X) = 0$ and is of general type.
\end{enumerate}
\end{prop}
\begin{proof}
\eqref{propitem:FNsurface sc1} These blow up surfaces are simply connected by birational invariance of  $\pi_1$.
For $k=3$ we  deal with $X = \mathbb{P}^2$ that has $\conFN(\mathbb{P}^2) = 3$ because of the ample line bundle ${\mathcal O}(1)$.

\smallskip

For $k = 2$ we consider the blow up $X \to \mathbb{P}^2$ in one point. This is the Hirzebruch surface $\mathbb{P}({\mathcal O} \oplus {\mathcal O}(-1))$, and $\Pic(X)$ is generated by the class of a fiber $F$ and the class of a section $S$ with self intersection $(S^2) = -1$. A divisor $L = aS + bF$ is nef if and only if $b \geq a \geq 0$. Consequently, a divisor $L = aS+bF$ is ample, i.e.\ in the interior of the nef cone, if and only if $b > a > 0$.  Moreover, as Hirzebruch surfaces are toric varieties, $aS + bF$ is globally generated if and only it is nef, see
\cite[Theorem 3.1]{mustata_vanishing_2002}. The canonical class is $K = -2S - 3F$. For ample divisors $L_i = a_i S + b_i F$, $i=1,\ldots, m$, we find that
\[
K + \sum_{i=1}^m L_i = (-2 + \sum_{i=1}^m a_i )S + (-3 + \sum_{i=1}^m b_i ) F.
\]
For $L_i = S + 2F$ this becomes $(m-2)S + (2m-3)F$, and that is nef, hence globally generated, if and only if  $m \geq 2$. When $m \geq 2$, for general ample divisor $L_i$, then
\[
-3 + \sum_{i=1}^m b_i \geq -3 + \sum_{i=1}^m (a_i+1) > -2  + \sum_{i=1}^m a_i  \geq 0,
\]
and so the corresponding adjoint divisor is globally generated. This shows that $\conFN(X) = 2$.

\smallskip

For $k = 1$ we consider the blow up $X \to \mathbb{P}^2$ in two points. This is a del Pezzo surface of degree $7$,
and $\Pic(X)$ is generated by the pullback $H$ of the line and the two exceptional fibers $E_1$ and $E_2$. A divisor $L = dH - a_1 E_1 -a_2E_2$ is nef,  if and only if
\[
d \geq a_1 + a_2 \quad  \text{ and}  \quad a_i \geq 0, \text{ for $i = 1,2$}.
\]
Consequently, $L$ being ample means that all inequalities are strict. Since we may think of $X$ as being the blow up in two torus invariant points, $X$ is a toric variety and so again by \cite[Theorem 3.1]{mustata_vanishing_2002} a divisor $L$ is globally generated if and only if $L$ is nef. The canonical divisor is $K = -3H + E_1 + E_2$ and not globally generated, hence $\conFN(X) \geq 1$. For any ample divisor $L = dH - a_1E_1 - a_2 E_2$, i.e.\ $d \geq a_1 + a_2 + 1$ and $a_i \geq 1$, the adjoint divisor
\[
K + L = (d-3)H - (a_1-1)E_1 - (a_2-1)E_2
\]
has $(a_i-1) \geq 0$ and
\[
d-3 \geq a_1 + a_2 + 1 - 3  = (a_1 - 1) + (a_2 - 1),
\]
hence $K+L$ is globally generated. This shows that $\conFN(X) = 1$.

\smallskip

\eqref{propitem:FNsurface sc2}  The hyperplane is simply connected due to Lefschetz hyperplane theorem for the fundamental group. By Noether-Lefschetz \cite{lefschetz_certain_1921} and degree $d \geq 4$ (see also \cite[Exp.~XIX Th\'eor\`eme 1.2]{deligne_groupes_1973}), the Picard group $\Pic(X)$ is generated by ${\mathcal O}(1)|_X$. This is the reason to restrict to very general hyperplanes. The convex Fujita number is then calculated as in the proof of \pref{prop:FNcompleteintersection}. Since the degree is at least $d \geq 5$, the canonical bundle $\omega_X = {\mathcal O}(d-4)|_X$ is very ample.
\end{proof}

\subsection{Convex Fujita numbers for simply connected varieties}

We now prove \tref{thmABC:FNsimplyconnected}.

\begin{thm}
\label{thm:FNsimplyconnected}
Let  $n \geq 2$ be an integer, and let $0 \leq k \leq n+1$. There is a simply connected smooth projective variety $X$ of dimension $n$ with convex Fujita number $\conFN(X) = k$. Moreover, if $k=0$, then $X$ can be chosen to be of general type.
\end{thm}
\begin{proof}
We remark first that $n$-dimensional complete intersections in $\mathbb{P}^{n+r}$ are simply connected by the Lefschetz hyperplane theorem as long as $n \geq 2$. For $n \geq 3$ thus the proof reduces to \pref{prop:FNcompleteintersection}, because all values of $k$ can be obtained, e.g. by $r = 1$ and $d_1 = n+2-k$. If $k=0$, then we choose $r=1$ and $d_1 > n+2$ which forces $\omega_X$ to be very ample and $X$ to be of general type.

The case $n=2$ is nothing but \pref{prop:FNsurfaces simply connected}.
\end{proof}

\section{Varieties with prescribed fundamental group}
\label{sec:FujitaSimple}

In this section we construct smooth projective varieties with a given fundamental group and varying dimension and convex Fujita number.

\subsection{Fujita simple surfaces with prescribed fundamental group}

We now prove the surface case of \tref{thmABC:FNsimple}.

\begin{prop}
\label{prop:FNsimpleSurface}
Let $\pi$ be a projective group. Then there is a smooth projective surface $X$ of general type with fundamental group isomorphic to $\pi$ and convex Fujita number $\conFN(X)= 0$.
\end{prop}
\begin{proof}
Let $Y$ be a smooth projective variety with $\pi_1(Y) = \pi$. Upon replacing $Y$ by its product with some projective space, we may assume $\dim(Y) \geq 3$.  Passing to a smooth complete intersection of $\dim(Y)-3$ very ample divisors if necessary, we assume going forward that $\dim(Y) = 3$.

Fixing an ample divisor $H$ on $Y$ and an integer $p \geq 5$ such that $K_Y + pH$ is ample and globally generated, we choose $X$ to be a very general smooth hypersurface in the linear system $|pH|$.  By the Lefschetz hyperplane theorem, $\pi_1(X)$ is isomorphic to $\pi$. By Noether-Lefschetz for arbitrary $3$-folds, see \cite{joshi_noether-lefschetz_1995} and more effectively using global generation of $K_Y + pH$ by \cite[Theorem~1]{ravindra_girivaru_v_noetherlefschetz_2009},
we have
\[
\Pic(X) = \Pic(Y)
\]
via restriction. This means that for all $\mathscr{L}_1, \mathscr{L}_2 \in \Pic(X)$ there are extensions $\mathscr{M}_1, \mathscr{M}_2 \in \Pic(Y)$ with $\mathscr{L}_i = \mathscr{M}_i|_X$ and thus
\[
(\mathscr{L}_1 \cdot  \mathscr{L}_2)_X = (\mathscr{M}_1 \cdot  \mathscr{M}_2 \cdot {\mathcal O}_Y(pH))_Y \in p{\mathbb Z}
\]
is divisible by $p$.  This shows $\conFN(X) \leq 1$ by \pref{prop:FNsurfacesReider divisible intersection numbers}.

By adjunction, the canonical bundle of $X$ is $\omega_X = (\omega_Y \otimes {\mathcal O}_Y(pH))|_X$. Since $\omega_Y \otimes {\mathcal O}_Y(pH)$ is ample and globally generated, its restriction to $X$ is also ample and globally generated, so $\conFN(X) = 0$ as desired.
\end{proof}

 \tref{thmABC:FNsimple} now follows easily with the exception of dimension $n=3$. Indeed, given a projective group $\pi$ we first choose a Fujita simple surface $S$ of general type with $\pi_1(S) = \pi$ as in \pref{prop:FNsimpleSurface}. If $n=2$ we are done. Otherwise by \tref{thmABC:FNsimplyconnected}, since we excluded $n=3$, we have a complete intersection $Y$ of general type and dimension $n-2$ which is Fujita simple and simply connected. By  \pref{propABC:FNproductwithPn} the variety $X = S \times Y$ is Fujita simple, it is of general type and dimension $n$, and with $\pi_1(X) = \pi$ by the K\"unneth formula for $\pi_1$. This constructs the variety asked for in  \tref{thmABC:FNsimple}.

\subsection{Totally branched cyclic covers}

It remains to construct a threefold that satisfies the needs of \tref{thmABC:FNsimple}. Since the construction is not particularly different in all dimensions $n \geq 3$, we do not specialize in threefolds now. We dealt with surfaces first in \pref{prop:FNsimpleSurface} because we wanted to highlight how far one gets by only invoking Reider's method instead of the analytic methods of Angehrn-Siu \cite{angehrn_effective_1995}.

\begin{rmk}
We first recall the construction of the cyclic and totally branched covering $X \to Y$ of a smooth projective variety $Y$ with respect to a line bundle $\mathscr{L} \in \Pic(Y)$, a degree $d \geq 1$, and a smooth divisor $B$ (branch locus) in the linear system associated to $\mathscr{L}^{\otimes d}$, more precisely to an isomorphism
\[
s : \mathscr{L}^{\otimes d} \xrightarrow{\sim} {\mathcal O}_Y(B).
\]
The $\mu_d$-torsor $X \to Y$ is constructed as the relative spectrum
\[
f: X = \Spec_Y\big(\Sym^\bullet(\mathscr{L}^{-1})/\mathscr{L}^{\otimes -d} = \mathscr{O}_Y(-B)\big) \to Y.
\]
which, locally with an equation $\{s=0\}$ for the Cartier divisor $B$, solves the equation $t^d = s$ inside the line bundle $\mathscr{L}$ globally. The variety $X$ is again smooth projective by Abhyankar's lemma, and also $R = f^{-1}(B)_{\rm red}$ (ramification locus) is smooth, in fact isomorphic to $B$. We have the well known relations
\begin{align*}
	\mathscr{O}_X(R) & \simeq f^\ast \mathscr{L}, \\
	f^\ast B & =  d \cdot R, \\
	\omega_X(R) & \simeq f^\ast (\omega_Y(B)).
\end{align*}
It follows that
\begin{equation}
	\label{eq:ramifiedcanonical bundle formula}
	\omega_X \simeq f^\ast(\omega_Y(B)) \otimes {\mathcal O}_X(-R)  \simeq f^\ast \big(\omega_Y \otimes \mathscr{L}^{\otimes (d-1)}\big).
\end{equation}
\end{rmk}

The next lemmas record the effect of various Lefschetz theorems applied to functors evaluated at the following diagram showing the cyclic totally branched covering constructed above.
\begin{equation}
	\label{eq:branchedcovering}
	\xymatrix@M+1ex{
		R \ar@{^(->}[r] \ar[d]^{f|_R} \ar[d]_{\simeq} & X \ar[d]^f \\
		B \ar@{^(->}[r] & Y.
	}
\end{equation}

\begin{lem}
	\label{lem:pi1 in branched cover}
	Let $f: X \to Y$ be the branched $\mu_d$-cover constructed above associated to $\mathscr{L}$ and $B$. If $\mathscr{L}$ is ample and the dimension $\dim(Y)$ is at least $3$, then $f$ induces an isomorphism
	\[
	f_\ast : \pi_1(X) \xrightarrow{\sim} \pi_1(Y).
	\]
\end{lem}
\begin{proof}
	Since $\mathscr{L}$ is ample, both $B$ in $Y$ and $R$ in $X$ are ample. By the Lefschetz hyperplane theorem, the functor $\pi_1$ applied to diagram \eqref{eq:branchedcovering} has isomorphisms for all but the map induced by  $f$. Hence also $f_\ast$ is an isomorphism.
\end{proof}

\begin{rmk}
\lref{lem:pi1 in branched cover} also holds in case $\dim(Y) = 2$ and then with the weaker assumption that $B^2 > 0$. This is proven in \cite[Proposition 1]{kulikov_numerically_2014} based on a result by Nori.
\end{rmk}

\begin{lem}
	\label{lem:Pic in branched cover}
	Let $f: X \to Y$ be the branched $\mu_d$-cover constructed above associated to $\mathscr{L}$ and $B$. If $\mathscr{L}$ is ample and the dimension $\dim(Y)$ is at least $4$, then $f$ induces an isomorphism
	\[
	f^\ast : \Pic(Y) \xrightarrow{\sim} \Pic(X).
	\]
Moreover, if $d \gg 0$ is sufficiently large and in addition $B$ is chosen to be very general, then the same conclusion holds when $\dim(Y) \geq 3$.
\end{lem}
\begin{proof}
Again, since $\mathscr{L}$ is ample, both $B$ in $Y$ and $R$ in $X$ are ample.

Let first $\dim(Y)$ be at least $4$. By the Lefschetz hyperplane theorem for $\Pic(-)$,
see \cite[Exp.~XII Cor.~3.6]{grothendieck_cohomologie_1965}, the functor $\Pic$ applied to diagram \eqref{eq:branchedcovering} has isomorphisms for all but the map induced by  $f$. Hence also $f^\ast$ is an isomorphism.

For $\dim(Y) = 3$ and very general $B$ we use again the Noether-Lefschetz theorem of \cite{joshi_noether-lefschetz_1995} to deduce that the restriction $\Pic(Y) \to \Pic(B)$ is an isomorphism. We cannot use the same argument for the restriction $\Pic(X) \to \Pic(R)$, since we do not control whether $R$ is very general in $X$, and most likely it is not. But by the Lefschetz hyperplane theorem for $\Pic(-)$,
see \cite[Exp.~XII Cor.~3.6]{grothendieck_cohomologie_1965}, the restriction $\Pic(X) \hookrightarrow \Pic(R)$ is still injective, and that is sufficient to conclude as in the proof before when $\dim(Y) \geq 4$.
\end{proof}

\begin{prop}
	\label{prop:FNin branched cover}
	Let $f: X \to Y$ be the branched $\mu_d$-cover constructed above associated to an ample line bundle $\mathscr{L}$ and a smooth divisor $B$. We assume that the dimension $\dim(Y)$ is at least $3$, where if $\dim(Y) = 3$ we ask $d \gg 0$ to be large and $B$ to be very general. Then the convex Fujita number is bounded as follows:
	\[
	\conFN(X) \leq \max\{0, \conFN(Y) + 1 - d \}.
	\]
Moreover,  if $d - 2 \geq \conFN(Y)$, then $\omega_X$ is ample and globally generated.
\end{prop}
\begin{proof}
	Let $\mathscr{L}_1, \ldots, \mathscr{L}_s$ be ample line bundles on $Y$. Due to Lemma~\ref{lem:Pic in branched cover} there are line bundles $\mathscr{M}_i$ on $X$ with $\mathscr{L}_i = f^\ast \mathscr{M}_i$. Moreover, as $f$ is finite, the line bundles $\mathscr{M}_i$ are also  ample. Since
	\[
	\omega_X \otimes (\mathscr{L}_1 \otimes \ldots \otimes \mathscr{L}_s) \simeq f^\ast \big(\omega_Y \otimes \mathscr{L}^{\otimes (d-1)} \otimes  \mathscr{M}_1 \otimes \ldots \otimes \mathscr{M}_s \big),
	\]
	and because being globally generated pulls back under morphisms, the left hand side is globally generated as soon as $s + d - 1 \geq \conFN(Y)$. The estimate for $\conFN(X)$ follows.

If $d - 2 \geq \conFN(X)$, then $\omega_Y \otimes \mathscr{L}^{\otimes (d-2)}$ is globally generated. The tensor product of an ample line bundle $\mathscr{L}$ with a globally generated line bundle $\omega_Y \otimes \mathscr{L}^{\otimes (d-2)}$ is again ample, hence $\omega_X$ is ample as the pull back of  $\omega_Y \otimes \mathscr{L}^{\otimes (d-1)}$.
\end{proof}

\subsection{Fujita simple varieties with prescribed fundamental group and dimension}

The proof of \tref{thmABC:FNsimple} begun in \pref{prop:FNsimpleSurface} will now be completed by the following proposition.

\begin{prop}
\label{prop:FNsimpleVariety}
Let $\pi$ be a projective group and $n \geq 3$. Then there is a smooth projective variety $X$ of general type and dimension $n$ with fundamental group isomorphic to $\pi$ and convex Fujita number $\conFN(X)= 0$.
\end{prop}
\begin{proof}
We can argue as in the proof of the surface case \pref{prop:FNsimpleSurface} that there is a smooth projective variety $Y$ of general type with $\pi_1(Y) = \pi$ and dimension $n$. Now we choose an ample line bundle $\mathscr{L}$ on $Y$.

By \pref{prop:FNisFinite}, the convex Fujita number $\conFN(Y)$ is finite. We choose $d \geq \conFN(Y) + 2$ large enough such that by Bertini we find a smooth divisor $B$ in the linear system associated to $\mathscr{L}^{\otimes d}$. Let $f: X \to Y$ be the branched $\mu_d$-cover constructed above associated to $\mathscr{L}$ and $B$.  If $\dim(Y) = 3$ we moreover ask $d \gg 0$ to be sufficiently large and $B$ very general, so that the conclusion of
\lref{lem:Pic in branched cover} holds. Then \pref{prop:FNin branched cover} shows that $X$ is Fujita simple and of general type.
\end{proof}

\subsection{Convex Fujita numbers with prescribed fundamental group and dimension}
We now aim to prove \tref{thmABC:anyFNanyPi1} of the introduction.

\begin{thm}
\label{thm:anyFNanyPi1}
Let $\pi$ be a projective group, and let $n \geq 2$ be an integer, and let $0 \leq k \leq n-1$. There is a smooth connected projective variety $X$ of dimension $n$ with the following properties:
\begin{enumerate}[label=(\roman*),align=left,labelindent=0pt,leftmargin=*,widest = (iii)]
\item
$\pi_1(X)$ is isomorphic to $\pi$, and
\item
$\conFN(X) = k$.
\end{enumerate}
\end{thm}
\begin{proof}
The case $k=0$ is the content of  \tref{thmABC:FNsimple}, so that case is done. Let $S$ be such a Fujita simple surface with $\pi_1(S) = \pi$.

Let first $n$ be at least $4$. If $0 \leq k \leq n-1$, then there exists a simply connected variety $Y$ of dimension $n-2 \geq 2$ and convex Fujita number $k$ by \tref{thmABC:FNsimplyconnected}. By the K\"unneth formula the product $X = S \times Y$ has $\pi_1(X) = \pi_1(S) = \pi$, and, since $\rH^{1}(Y,{\mathcal O}_{Y})=0$ for the simply connected variety $Y$, we have from \cref{cor:FNproductwithPn} that $\conFN(X) = \max\{0,k\} = k$.

If $n = 3$ and $k=2$, then we can take $Y = \mathbb{P}^1$ with $\conFN(Y) = 2$ and again conclude that $X = S \times Y$ has the required fundamental group and convex Fujita number $2$.

The only cases missing now are the case $(n,k) = (2,1)$ and $(3,1)$ which we deal with separately in \pref{prop:FNn2k1} and \pref{prop:FNn3k1} below.
\end{proof}

\begin{prop}
\label{prop:FNn2k1}
Let $\pi$ be a projective group. There exists a smooth projective surface $S$ such that the blow-up $S' \to S$ in a point yields a smooth projective surface with $\conFN(S') = 1$ and $\pi_1(S') =\pi$.
\end{prop}
\begin{proof}
We can argue as in the proof of  \pref{prop:FNsimpleSurface} that there is a smooth projective variety $Y$ of general type with $\pi_1(Y) = \pi$ and dimension $3$. Now we choose a very ample line bundle ${\mathcal O}(1)$ on $Y$ and choose $S$ a very general smooth divisor in the linear system of ${\mathcal O}(24)$ by Bertini's theorem. Upon replacing ${\mathcal O}(1)$ by a multiple initially, we may assume by Noether-Lefschetz, see \cite{joshi_noether-lefschetz_1995}, that $\Pic(S) = \Pic(Y)$. The choice of the power $24$ then forces all intersection numbers of line bundles on $S$ to be divisible by $24$.

By the Lefschetz hyperplane theorem and the birational invariance of the fundamental group the blow up $\sigma: S' \to S$ in a choice of a point $P$ on $S$ is a smooth projective surface $S'$ with
\[
\pi_1(S') = \pi_1(S) = \pi_1(Y) = \pi.
\]

Since $S'$ is not minimal the canonical divisor $K_{S'}$ is not nef and a fortiori not globally generated. Thus $\conFN(S') \geq 1$. It remains to show that for all ample divisors $L$ on $S'$ the adjoint divisor $K_{S'} + L$ is globally generated.

Since $\Pic(S')$ equals $\Pic(S) \oplus {\mathbb Z}$, with the summand ${\mathbb Z}$ spanned by the exceptional divisor $E = \sigma^{-1}(P)$, we find that $L = \sigma^\ast M -aE$ for some divisor $M$ on $S$  and $a \in {\mathbb Z}$. We compute
\[
(L^2) = (M^2) - a^2 > 0,
\]
so modulo $24$ it's the negative of a square. Squares modulo $24$ are $0,1,4,9,12,16$, hence
\[
(L^2) \geq 8.
\]
Reider's theorem \cite[Theorem 1 (i)]{reider_vector_1988} implies that $K_{S'}  + L$ is globally generated unless we are in the exceptional case: there is an effective divisor $C'$ on $S'$ such that $(C'^2) = 0$ and $(L \cdot C') = 1$. Since $L$ is ample and $C'$ is effective, it follows from $(L \cdot  C') = 1$  that $C'$ must be irreducible and reduced. As $(E^2) = -1$ we can exclude $C' = E$. Hence $C'$ is the strict transform of an irreducible and reduced curve $C$ on $S$. Let $m_C$ be the multiplicity of $C$ in $P$. Then
\[
0 = (C'^2) = (C^2) - m_C^2 \equiv -m_C^2  \pmod{24}.
\]
It follows that $12$ divides $m_C$. On the other hand, we have $(M \cdot C) \equiv 0 \pmod{24}$ and so
\[
1 = (L \cdot C') =  (M \cdot C) - a m_C \equiv 0 \pmod{12},
\]
a contradiction.
\end{proof}

\begin{prop}
\label{prop:FNn3k1}
Let $\pi$ be a projective group, and let $S$ be a smooth projective surface with $\conFN(S) \leq 2$. Let ${\mathcal M}$ be a very ample line bundle on $S$, and set $\mathscr{L} = {\mathcal M} \boxtimes {\mathcal O}(1)$ on $Y = S \times \mathbb{P}^1$. Then the double cover $f:X \to Y$ branched in a very general smooth divisor $B$ of the linear system associated to $\mathscr{L}^{\otimes 2}$ is a smooth projective threefold with $\conFN(X) = 1$ and $\pi_1(X) =\pi$.
\end{prop}
\begin{proof}
Let $\text{pr}: Y \to S$ be the projection map. Since $\conFN(S) \leq 2$, the line bundle
\[
\omega_Y(B) = \big(\omega_S \otimes \mathscr{M}^{\otimes 2}\big) \boxtimes \big(\omega_{\mathbb{P}^1} (2)\big) = \text{pr}^\ast(\omega_S \otimes \mathscr{M}^{\otimes 2})
\]
is globally generated. Therefore the  effective version of the Noether-Lefschetz theorem proved in
\cite[Theorem~1]{ravindra_girivaru_v_noetherlefschetz_2009} applied to ${\mathcal O}_Y(1) \coloneq \mathscr{L}^{\otimes 2}$ shows that the restriction map
\[
\Pic(Y) \xrightarrow{\sim} \Pic(B)
\]
is an isomorphism. This suffices to show as in \lref{lem:Pic in branched cover} that
\[
f^\ast : \Pic(Y) \to \Pic(X)
\]
is an isomorphism. By \pref{propABC:FNproductwithPn} we have $\conFN(Y) = 2$, and by
 \pref{prop:FNin branched cover} we have $\conFN(X) \leq 1$. It remains to show that $\omega_X$ is not globally generated. We compute its global sections using $f_\ast {\mathcal O}_X = {\mathcal O}_Y \oplus \mathscr{L}^{-1}$ as
 \[
 \rH^0(X,\omega_X) = \rH^0(X, f^\ast(\omega_Y \otimes \mathscr{L})) = \rH^0(Y, \omega_Y \otimes \mathscr{L}) \oplus \rH^0(Y,\omega_Y).
 \]
 Furthermore, by the K\"unneth formula and the product structure of $Y$ and $\omega_Y \otimes \mathscr{L}$
 \begin{align*}
 \rH^0(Y, \omega_Y \otimes \mathscr{L}) & = \rH^0(S,\omega_S \otimes \mathscr{M}) \otimes \rH^0(\mathbb{P}^1,{\mathcal O}(-1)) = 0, \\
  \rH^0(Y,\omega_Y) & = \rH^0(S,\omega_S) \otimes \rH^0(\mathbb{P}^1,{\mathcal O}(-2)) = 0,
 \end{align*}
 both vanish. Hence $\omega_X$ has no global sections and the proof is complete.
\end{proof}

\begin{bibdiv}
\begin{biblist}

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      author={Siu, Yum~Tong},
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}

\bib{bauer_tensor_1996}{article}{
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}

\bib{ein_syzygies_1993}{article}{
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}

\bib{fujita_polarized_1987}{article}{
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}

\bib{ghidelli_logarithmic_2021}{misc}{
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      author={Lacini, Justin},
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}

\bib{heier_effective_2002}{article}{
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}

\bib{helmke_fujitas_1997}{article}{
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\bib{joshi_noether-lefschetz_1995}{article}{
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\bib{kawamata_fujitas_1997}{article}{
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\bib{kulikov_numerically_2014}{article}{
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\bib{laterveer_linear_1996}{article}{
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}

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\end{biblist}
\end{bibdiv}

\end{document}

redpajama_set_name: RedPajamaArXiv


text:
Sheffield Theatres announce Coriolanus cast
on 5th February 2020 at 10:28 am
The complete cast of Sheffield Theatres' upcoming production of Coriolanus has been announced. Under the direction of Robert Hastie, who received critical acclaim for his production of Julius Caesar in 2017, the adaption is set to generate considerable excitement as a contemporary reading of the classic tragedy. The performance stars Tom Bateman as the tragic hero, with Hermon Berhane as Virgilia and Malcom Sinclair as Menenius Agrippa. Sheffield Theatre's intergenerational company, Sheffield People's Theatre, complete the cast.
Running at the Crucible Theatre from Friday 6th – Saturday 28th March, the play is based on the life of Caius Marcius Coriolanus, a Roman military commander who seeks power following his defeat of the Volscian army. In timeless Shakespearian style, the play chronicles Coriolanus' fall from political power after losing the favour of his public, and his last-ditch attempt at to seize power as he condemns democratic rule and the city he once championed.
Tickets for the performance are available now from the Sheffield Theatre's Box Office, online at sheffieldtheatres.co.uk or on 01142496000.
CultureWhat's On
Plans in for a Ninja Warrior adventure park in Sheffield
ShAFF returns - with a rebellious streak!


text:
'More safety measures needed at Bow roundabout' - Assembly Member
Robin De Peyer
Published: 11:00 AM July 4, 2013 Updated: 8:37 AM October 14, 2020
Traffic at the busy Bow roundabout - Credit: Archant
Continuing fears over the safety of cyclists and pedestrians at notorious black spot Bow roundabout should be tackled urgently, it has been urged.
London Assembly Member for City and East John Biggs welcomed the ongoing work in the south west corner of the busy junction, between Bow and Stratford, which is designed to further improve cyclists' safety.
The works are part of a series of measures implemented since the death of two cyclists in just three weeks in 2011.
But the Tower Hamlets mayoral candidate for Labour urged Transport for London (TfL) to continue improving safety at the junction, with greater emphasis on pedestrians.
"I welcome the long overdue safety measures which are now underway for cyclists travelling from west to east," he said.
"However, I am still very concerned about safety - particularly of pedestrians - at Bow roundabout.
"TfL say motorists will be faced with mile long queues should pedestrian crossings be installed. Whilst we don't want life to be harder for drivers, pedestrian safety is extremely important and something must be done to tackle this urgent issue."


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text:
+ Exhibitor log in
2023 application
Apply for vip
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Colour Senses Project
3430 NW 7 Avenue
FL 33127 Miami
Email : coloursenses@gmail.com
URL : www..coloursensesproject.com
German Chaves
Caterina Mejia
The Colour Senses Projec tIs an art collective founded in the heart of Allapattah located in the city of Miami It is dedicated to promote an artistic movement in the area with programs created specifically for this purpose so they can have positive impact.
Hernando Alzate SAKIKO (Blossom Girl
Cate Karate Money Laundry. Series
Valentina Bilbao The Point of Encounter
Iurii Poniatovskii Gaddi Torso
Jon Gilchrist Born Again
Julia Dogan Persistence
Nelson Babilonia Contra la Corriente
Leonor Anthony The Last Supper
Fabia Nitti Wave
Felton Weller Untitled
Alina Shimova Free Mind
Charly Silva Serpentine
I ONCE FOUND A SMALL ACTION FIGURE LYING ON THE FLOOR. I PICKED IT UP AND FELT I WAS RESCUING A SURVIVOR WHO HAD MADE IT THROUGH A VERY DIFFICULT LIFE. THAT DAY I WAS INSPIRED TO LOOK FOR A WAY TO COMBINE MANY OF THESE FORGOTTEN TOYS INTO A SCULPTURAL WORK AND CREATE NEW ICONS, SHAPED BY HUNDREDS OF DIFFERENT STORIES COEXISTING IN HARMONY AND BREATHING LIFE INTO A NEW BEING... A BEING THAT WILL HOPEFULLY LIVE MUCH LONGER.
I once found a small action figure lying on the floor. I picked it up and felt I
was rescuing a survivor who had made it through a very difficult life. That
day I was inspired to look for a way to combine many of these forgotten
toys into a sculptural work and create new icons, shaped by hundreds of
different stories coexisting in harmony and breathing life into a new
being ... a being that will hopefully live much longer.
H. Alzate
CateKarate is a contemporary artist based in Miami, FL. Her inspirations surge from her lifelong travels and multicultural background, her development in colors and techniques. Are mostly born in her momentarily presence.
CateKarate is a Visual artist based in Miami, FL. Her
inspirations surge from her travels and multicultural
background, her development in colors and techniques are
mostly born in her momentarily presence.
Cate born in Miami, FL in the late 1980s, raised in
Cali, Colombia. until she was 14 years old. Her family
then decided to move to Germany.
She finished her Bachelor of Arts in Photography in
Stuttgart.
After college, she went back to America. Where she started
painting in an abstract form of acrylic, a year later she
returned to Miami and did her first art exhibition. Painting a
lot with spray paint and mixed media.
At the moment she resides in Miami, inspiring her
homeland.
As one of her main projects is the Colour Senses Art
Collective Gallery helping emerging artists and the
community of Miami to surge in the light.
Valentina Bilbao's art is gently encapsulating, where such harmonious movement
Appears to have an alluring effect, almost as if it were to pull the viewer into the canvas.
The inspiration for the motion and color in each of her paintings is a reflection of the
Music which she listens to while painting, specific to each individual piece.
Stunning piece of Hellenistic art. A fragment of the famous plot of the taming of the centaur by Cupid. A plasticity masterpiece that inspired the young Michelangelo. The original is displayed in the Uffizi Gallery. Florence.
Jon Gilchrist is a self-taught abstract artist born in 1975 in Oklahoma City. Later in life Jon was exposed to contemporary art and found himself more drawn to abstract art, due to his introduction to artists such as Joan Mitchell, Mark Rothko, Sam Francis, Robert Motherwell, Norman Lewis, and Romare Bearden. Jon mostly expresses these influences through his use of texture, materials, arrangement, application, and elevating his own brand of social commentary to a more universal stature. Painting allows Jon to share his story, and his soul, in hopes that this strikes a nerve – that it resonates.
Any goal is achievable if you persevere in achieving it. I depicted a woman on the hood of the Ferrari car, and there is no coincidence.
Ferrari has the symbol of a horse, at least that's what the company says, and in my work it symbolizes a horse. In some cultures there is such an expression - "to ride the horse", it means to be at the top of the game, to become a winner, to get what you want.
And Persistence is the key to success.
My paintings are inspired by my dreams; I do not choose dreams, nor do I choose what to paint. I create with symbols, lines, colors, canvas on canvas juxtaposing composition. Painting or drawing in a style of automatism, unleashes my imagination expressing emotions and desires, allowing me to create the unknown; The way I see my artworks is a response from my live experience that projected into a poetic theme.
Comprised of 13 individual and unique pieces inspired by Leonardo Da Vinci's Last Supper mural, painted during the Italian High Renaissance dated 1495 -1498.
Each sculpture represents our last supper should we not act immediately and do something about Climate change and the rising water level.
In Da Vinci's, Last Supper , he captures the moment in which Christ tells his apostles "Truly, truly I say to you, one of you will betray me "
In our case, all of us have betrayed our beautiful planet and it will take all of us united, not only to undo all the damage we have created but to help each other through the consequences of our actions.
Each piece has a different color acrylic signifying the different colors of the oceans, including the receding Ice cap. The Jesus figure is in gold.
The materials used in each sculpture are recycled and reused materials.
The entire 13 pieces will be in the MuFe Museum of Contemporary Art Permanent Collection in Cancun Mexico, starting in May 2023.
Fabia Nitti Expresses her quest for life in simple geometric shapes with complex questions surrounding them. One of her sculptures, the Square? Transforms the artist into a superimposed sequence based on real life, due to the perfection of the square shape. Day to day life experiences are shown to form episodes of life.
In this artwork I wanted to transmit the inseparable connection between nature and human being, the connection of the energy of the earth with the energy of woman. They intersect like neural networks in our minds. And this common energy gives birth and creates the world around us.
Large-scale aluminum sculpture with simple, linear forms. Minimalistic and interactive with the public in its intimate and soundproof space, where the wind converts to the serpentine. One of the sculptures by Charlie Silva; is part of the project in the Dante Facell Park in South Miami, Florida.
+ Hernando Alzate (+)
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https://www.fairplanet.org/story/restructuring-nigeria-what-will-work-part-2/
Restructuring Nigeria: What will work? (Part 2)
Humans, Economy
Kolawole Talabi
anti-corruption, Aso Rock Villa, Boko Haram, Niger-Delta
Not satisfied, it looks beyond the horizon across the length and breadth of the country. Beneath the majestic façade of this huge monolith lies the nucleus of the Nigerian state ― the Villa, the seat of government.
In the years since the restoration of democratic rule in 1999, Nigerian heads of state have found themselves clothed with authority akin to those of Rome's first-century emperors. Buoyed by an oversized bureaucracy, an elected statesman can easily imagine himself a royal in the corridors of Aso Rock Villa. The Nigerian presidency essentially commands the power of life and death over a nation of 180 million people. But the citizenry are fed up with Abuja's incompetence. The current resident of the Villa rode on the wave of anti-corruption but that message has since been lost to the economic recession that rocked the country and the mismanagement of public resources from 2015 onwards. Now, more and more Nigerians want Abuja to cut its coat. The refrain in some parts of society seems to be 'Devolution or else revolution!' Nura Maaji, an academic at the Bayero University, Kano and a prominent figure in the Civil Society Legislative Advocacy Centre (CISLAC) talks to fairplanet on why he wants to see Nigeria re-federalized in the third and final interview in our series on Nigeria's 57th independence anniversary.
fairplanet: Many Nigerians are clamouring for true federalism in the hope that it will lead to good governance as power shifts from the centre in Abuja to the states of the federation. What are your thoughts on devolution?
Maaji: I think it's important for us to understand that nation building is a process involving series of negotiations. This means that there is no one perfect model that can be relied upon at all times, hence any model is subject to further change to suit the thinking and aspirations of the society based on the circumstances it finds itself per time. To this end, devolution of power involves an act of decentralization from the core, or the capital to the constituent units for the purpose of giving the subnational divisions the wherewithal to cater for some certain aspects of governance without the intervention of the centre. Nigeria really needs to be re-federalized or restructured in order to devolve some powers from Abuja to the 36 states. This is because the centre has been abusing the powers allotted to it. Though, the northern part will be in a disadvantaged position if such exercise occurred.
The poorest states in Nigeria are in the northern part of the country where you hail from. The elites from the north have been accused of favouring the status quo, a situation that allows corrupt politicians to loot the central treasury. Is the north slowing down Nigeria's progress?
Northern elites, particularly politicians, are the most unpatriotic class of elites I have ever seen. They are the greatest impediment to the development of the region. Unless our youths call them to order, the north will continue to be described as the region delaying the country's progress. They have looted our taxes and spread poverty as a mechanism of political control.
The Niger Delta is essentially the cash cow of the country. In the event that the South-South region secedes, how do you suppose the north will survive economically?
The north is not a poor region, but one left with untapped potentials for greatness. Tell me, what happened before oil exploration in Nigeria? The revenues realised from the export of groundnut grown in the north was used in running the country. In fact, those funds helped a great deal in the investment that led to the exploration of oil in the Niger Delta. This means that the present predicament in the north is the product of the moral irresponsibility of its political class.
The current administration has recorded considerable success in the fight against Boko Haram. But there is also the ongoing crisis in the Southeast. Is President Muhammadu Buhari handling the IPOB case properly?
I think this government is a huge disappointment in the way they have been handling the Nigeria project. While one can commend them for degrading the capacity of Boko Haram terrorists to carry out attacks, the reality is that there is still a long way to go in terms of ending the conflict. The problem is: government handle security issues in this country in a manner that allows vested interests to have a field day. I think the President is confused and he has no knowledge on how best to address the issue of corruption in the security sector. The IPOB issue is a product of the same trend.
If you were given the task of rewriting the constitution, what reforms would you incorporate into it?
I will remove the immunity clause for the political office holders, grant autonomy to local governments, separate the powers of the minister of justice from that of attorney general, separate the office of the accountant general of the federation from that of the accountant general of the federal government. I will also insist on a unicameral legislature rather than the unproductive and cumbersome bicameralism of our current National Assembly, among other things.
Describe briefly why your work as an advocate for social justice matters for the development of civil in Nigeria.
Nigeria is a very unequal society, quite lawless and extremely corrupt too. It is a society where a small percentage of its population uses state apparatuses to milk the country dry in the pursuit of their ill-conceived agendas. Between 1999 and 2014, Nigeria realised 75 trillion naira from oil according to a 2017 NEITI report, however, the government still fails to account for the whereabouts of those funds. We are a country where more than 80 million people are living below $1 per day in spite of the huge amount of revenues realised from oil and gas exports. I think there is no better place to advocate for social justice other than places like Nigeria, hence my resolve to face these monumental injustices.
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More and more Nigerians want Abuja to cut its coat. The refrain in some parts of society seems to be 'Devolution or else revolution!'
"I think it's important for us to understand that nation building is a process involving series of negotiations. This means that there is no one perfect model that can be relied upon at all times."
"Northern elites, particularly politicians, are the most unpatriotic class of elites I have ever seen."
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Please finish by going to "Manage Wallet", copy your blockchain key and save it. This key (think of it as a password) will allow you to access your wallet from any other device.
You will also need this key to recover your wallet in case you delete your browsing data as this is where your keys are stored. Click here to see how.
Please note: You'll get the free amount only once – to ensure that nobody claims the money twice you'll have to verify yourself with your phone number.
If you should experience any troubles contact hello@satoshipay.io, they will be happy to help.
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CookieSpy v6.0 Freeware Download - While surfing the Internet web sites your browser saves bits of information about your experience on the site (known as HTTP cookies, internet cookies, web cookies or simply cookies).
While surfing the Internet web sites your browser saves bits of information about your experience on the site (known as HTTP cookies, internet cookies, web cookies or simply cookies). They are your search history, some preferences, logins, and passwords. Information about data that web sites save locally in cookies may be important for you security. CookieSpy is a free utility application that displays HTTP cookies stored on your computer by all popular web browsers. It supports all major web browsers such as Internet Explorer, Mozilla Firefox, Google Chrome, Google Chrome Canary, Chromium, Safari, Opera, SeaMokey, Pale Moon, Comodo Dragon, Maxton and portable browsers. Cookies are displayed in a table view. You can easily sort and filter the cookie list, edit and delete cookies.
6.0 01/08/2015 Added support of Maxthon Nitro.
5.1 02/05/2014 Added a new context menu command to remove all cookies "Delete All".
Added a new context menu command for tab "Close other tabs".
Fixed a crash while trying to sort by "Domain" column.
Rearranged "Options" menu, added link to CookieSpy's blog.
5.0 09/29/2013 Added support of Yandex browser.
Added support of Opera 15 (and above) browser.
Added ability to protect a cookie from being deletededited in CookieSpy.
Displays information about known cookies.
Displays statistics of cookies usages.
4.3 04/18/2013 Allows you to delete specific cookies of Opera web browser.
Pirated Software Hurts Software Developers. Using CookieSpy Free Download crack, warez, password, serial numbers, torrent, keygen, registration codes, key generators is illegal. We do not host any torrent files or links of CookieSpy from depositfiles.com, rapidshare.com, any file sharing sites.


The Astro Zeppo is a stylistic pendant with a modern look. It features an oval shape with chrome detailing. Due to it;s design, the Zeppo would look great hanging in a dining room over a table or in a kitchen, hanging over an island. See the rest of the Astro's range here.


Breakfast is without a doubt my favourite meal. Personally, I feel it's important to have a good breakfast, it sets us up for the day. The children are going out to school and I really feel they need something substantial in their tummies before taking on the day ahead.
Now, I have spoken at length about breakfast here in the past, about my quest to get my family eating better breakfasts and not reaching for convenience. We are far from perfect, I do still buy Weetabix and cornflakes, Shredded Wheat too when it's on special offer, but I am pleased to say that our breakfast menu is much more varied and the cereal is not as frequent as pancakes or waffles, eggs, smoothies and now these breakfast bars.
My teen in particular really enjoys these, they are convenient but full of goodness and they keep her going until break time. They also freeze very well, which is very handy.
In a large bowl mix together the oats, flour, almonds, linseed and salt.
Put all of the wet ingredients into a separate bowl and whisk them together.
Pour the wet ingredients into the dry and mix together well. Everything should come together into one ball of what is an almost dough-like mix.
Place half of the mixture into the tin, you may find it easier to make a layer of small pieces that you can then push together. I find the easiest way to spread this mix is with a separate piece of parchment laid on it and then guided with the back of a spoon.
When the bottom layer is nice and smooth, place the jam on top and spread it out.
Then repeat the process with the second half of the dough. Don't worry about gaps or the jam seeping through, these result in delicious chewy parts!
Bake for 30 minutes and leave in the tin to cool.
Once cold remove from the tin and cut into 12 bars.
We have two of these for breakfast, but I must confess, I have been known to have one with a cup of coffee too! They are delicious. I'd love to hear what you think of them if you decide to bake them.
Thank you 🙂 It's such a treat when that morning cuppa doesn't go cold!


Now also available in non-illuminated versions on the XTR riflescopes, and on the Burris Signature HD spotting scope.
The SCR™ Mil reticle is ideal for precision competition use. With its extended illumination area, you can engage targets at long distances even in low light.