Timeline for Néron–Severi group of Abelian surfaces

6 events

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Sep 21, 2023 at 12:08	comment	added	Stormblessed		I’m interested in understanding if the formula holds over $\mathbb{Z}$. What about in the case when $E$ and $E’$ are not isogenous so the Picard rank should be 2.
Aug 28, 2023 at 20:50	comment	added	R. van Dobben de Bruyn		... but surely this was known well before my paper! Here is a simple argument over $\mathbf C$: then $\operatorname{NS}(X)$ is the image of $c_1 \colon \operatorname{Pic}(X)\to H^1(X,\mathbf Z)$ (up to a finite kernel in $\operatorname{NS}$), which by the Lefschetz (1,1)-theorem is the intersection $H^{1,1}(X)\cap H^2(X,\mathbf Z)$. Then use that $$H^{1,1}(X\times Y) \cong \bigoplus_{i,j=0}^1 H^{i,j}(X)\otimes H^{1-i,1-j}(Y),$$ so besides $H^{1,1}(X)\cap H^2(X,\mathbf Z)$ and $H^{1,1}(Y)\cap H^2(Y,\mathbf Z)$, you get something like $\operatorname{Hom}_{\mathbf Z\text{-HS}}(H^1(X),H^1(Y))$.
Aug 28, 2023 at 16:53	comment	added	Piotr Achinger		(Since you assumed isogenous, not isomorphic, to $E\times E'$, the formula maybe only works after tensoring with $\mathbb{Q}$.)
Aug 28, 2023 at 16:51	comment	added	Piotr Achinger		See Lemma 3.1 in the following reference for a more general fact: Remy van Dobben de Bruyn "A variety that cannot be dominated by one that lifts" Duke Math. J. 170.7, p. 1251–1289 (2021).
Aug 28, 2023 at 16:47	comment	added	Piotr Achinger		$\ldots\oplus \mathrm{Hom}(E, E')$
Aug 28, 2023 at 16:30	history	asked	Stormblessed	CC BY-SA 4.0