The polarized degree of irrationality of $K3$ surfaces
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916776264073216 |
|---|---|
| author | Moretti, Federico |
| author_facet | Moretti, Federico |
| contents | Given a polarized variety $(X,L)$, we construct and study projections of low degree $ X\dashrightarrow \mathbb{P}(H^0(L^\vee)) \dashrightarrow \mathbb P ^n $ using the associated kernel bundles. As an application, we can show that the degree of irrationality of a very general $(1,6)$ abelian surface, as well as that of a very general $K3$ surface of genus $6$ is $3$. We also give new upper bounds for $K3$ surfaces of any genus. Moreover, in the case of surfaces, this observation can be used to show that maps of the degree at most $d$ move in families. We study the family of projections of minimal degree of a very general $K3$ surface of genus $4,5,6$. As a different application of our construction, we exhibit new rational maps of low degree for some hyper-Kähler varieties, abelian varieties and Gushel--Mukai threefolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_07289 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The polarized degree of irrationality of $K3$ surfaces Moretti, Federico Algebraic Geometry Given a polarized variety $(X,L)$, we construct and study projections of low degree $ X\dashrightarrow \mathbb{P}(H^0(L^\vee)) \dashrightarrow \mathbb P ^n $ using the associated kernel bundles. As an application, we can show that the degree of irrationality of a very general $(1,6)$ abelian surface, as well as that of a very general $K3$ surface of genus $6$ is $3$. We also give new upper bounds for $K3$ surfaces of any genus. Moreover, in the case of surfaces, this observation can be used to show that maps of the degree at most $d$ move in families. We study the family of projections of minimal degree of a very general $K3$ surface of genus $4,5,6$. As a different application of our construction, we exhibit new rational maps of low degree for some hyper-Kähler varieties, abelian varieties and Gushel--Mukai threefolds. |
| title | The polarized degree of irrationality of $K3$ surfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2303.07289 |