Timeline for Lefschetz type theorems for big and nef divisors
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 17, 2017 at 12:03 | comment | added | user82886 | Indeed we can assume that the restriction map tensored by $\mathbb{Q}$ is an isomorphism. Thanks. | |
| Mar 17, 2017 at 12:02 | history | edited | user82886 | CC BY-SA 3.0 |
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| Mar 17, 2017 at 9:19 | comment | added | Bertie | By the way, in light of Jason Starr's comment below, it would be good to clarify whether you are assuming just that $\rho(X)=\rho(D)$, or that the map on Picard groups tensored with $\mathbf Q$ is an isomorphism. | |
| Mar 16, 2017 at 23:48 | comment | added | Bertie | I think it is easy to arrange examples where this map is not an isomorphism. (Let me know if I messed up.) For example let $X$ be the blowup of $\mathbf P^n$ along a line $L$ and a point $p$ not in $L$. Let $Q$ be a smooth quadric hypersurface meeting $L$ in 2 distinct points but not containing $p$, and let $D$ be the proper transform of $Q$ on $X$. Then both $X$ and $D$ have Picard number 3, $D$ is nef and big, but the exceptional divisor over $p$ restricts to $0$ in $\mathrm{Pic} \, D$. | |
| Mar 16, 2017 at 23:40 | comment | added | Jason Starr | I just realized that the linear system above is basepoint free, but it is not big. | |
| Mar 16, 2017 at 22:59 | comment | added | Jason Starr | Fix a linear subspace $\Lambda\subset \mathbb{P}^n$ of codimension $3$ (so $n\geq 3$). Denote by $\pi:\mathbb{P}^n\setminus \Lambda \to \Pi$ the linear projection to $\Pi\cong \mathbb{P}^2$. In the blowing up $\nu:X\to \mathbb{P}^n$ along $\Lambda$ with exceptional locus $E$, consider the linear system of $\nu^*\mathcal{O}(2)(-2\underline{E})$. It seems to me that this is a big and basepoint free linear system (I should double-check). Yet a general member $D$ of this linear system is a projective space bundle over a conic $C\subset \Pi$, so $\text{Pic}(X)$ has index $2$ in $\text{Pic}(D)$. | |
| Mar 16, 2017 at 22:46 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
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| Mar 16, 2017 at 19:47 | history | asked | user82886 | CC BY-SA 3.0 |