## Fano 3-fold of degree 4

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Let $X$ be the intersection of two quadrics in $P^5$. It is well known that the intermediate Jacobian $J(X)$ is isomorphic to $J(C)$ for a genus 2 curve, related to the pencil of quadrics whose base locus is $X$.

It seemed then natural to me to ask the following question:

Is there an explicit construction where $X$ is obtained as a smooth blow-up of $P^3$, or of a smooth quadric, or of a $P^2$ bundle over $P^1$, along a curve isomorphic to $C$?

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## 2 Answers

The projection from a line $L_0$ is a birational isomorphism of $X$ onto $P^3$. It decomposes as the blow-up of the line $L_0$ followed by the contraction of a surface swept by lines intersecting $L_0$ onto a curve of genus 2 in $P^3$.

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Thank you, this is the construction I was looking for. Can the normal sheaf of the line $L_o \subset X$ be both $\mathcal{O}\oplus \mathcal{O}$ and $\mathcal{O}(1)\oplus \mathcal{O}(-1)$? – IMeasy Mar 18 2011 at 11:47
This is probably known, but just to draw a line at the end of this post: I think that, at least generically, the normal bundle is $\mathcal{O}\oplus \mathcal{O}$ and it is sent onto the quadric surface $Q$ in $P^3$ in which $X$ is a divisor of type $(2,3)$. – IMeasy Mar 19 2011 at 12:07

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The answer is 'no'. By the Lefschetz hyperplane theorem, the second betti number $b_2$ of $X$ is 1, so in particular the Picard group of $X$ is isomorphic to $\mathbb{Z}$. Since blow-ups and $\mathbb{P}^k$-bundles have Picard number $\ge 2$, it follows that no such description exists.

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Yes, you're of course right.   Anyway my question may still stand in the following terms. There could be a birational model of $X$ (say a blow-up) that dominates both $X$ and one of the varieties I mentioned in my question.    – IMeasy Mar 16 2011 at 10:26
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