Bézout's theorem for abelian varieties
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911161755107328 |
|---|---|
| author | Debarre, Olivier Moonen, Ben |
| author_facet | Debarre, Olivier Moonen, Ben |
| contents | Let $X$, $Y$ be closed irreducible subvarieties of an absolutely simple abelian variety of dimension $g$ over a field. If $\dim(X) + \dim(Y) \le g$, we prove that the addition morphism $X \times Y \to X + Y$ is semismall. As a consequence, we deduce that if $\dim(X) + \dim(Y) \ge g$, the subvarieties $X$ and $Y$ must meet (Bézout's theorem). If we drop the assumption that the abelian variety is absolutely simple, we prove that Bézout's theorem still holds if $X$ satisfies a nondegeneracy condition. These results were previously known only in characteristic zero. Our proof of the semismallness statement is based on the theory of perverse sheaves: using results of Krämer and Weissauer, we prove that for perverse sheaves $K$ supported on $X$, and $L$ supported on $Y$, the convolution product $K * L$ is again perverse. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14940 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bézout's theorem for abelian varieties Debarre, Olivier Moonen, Ben Algebraic Geometry 14K05 (14F06, 14F20) Let $X$, $Y$ be closed irreducible subvarieties of an absolutely simple abelian variety of dimension $g$ over a field. If $\dim(X) + \dim(Y) \le g$, we prove that the addition morphism $X \times Y \to X + Y$ is semismall. As a consequence, we deduce that if $\dim(X) + \dim(Y) \ge g$, the subvarieties $X$ and $Y$ must meet (Bézout's theorem). If we drop the assumption that the abelian variety is absolutely simple, we prove that Bézout's theorem still holds if $X$ satisfies a nondegeneracy condition. These results were previously known only in characteristic zero. Our proof of the semismallness statement is based on the theory of perverse sheaves: using results of Krämer and Weissauer, we prove that for perverse sheaves $K$ supported on $X$, and $L$ supported on $Y$, the convolution product $K * L$ is again perverse. |
| title | Bézout's theorem for abelian varieties |
| topic | Algebraic Geometry 14K05 (14F06, 14F20) |
| url | https://arxiv.org/abs/2509.14940 |