Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914602988601344 |
|---|---|
| author | Patakfalvi, Zsolt |
| author_facet | Patakfalvi, Zsolt |
| contents | We show that if $f\colon X \to T$ is a surjective morphism between smooth projective varieties over an algebraically closed field $k$ of characteristic $p>0$ with geometrically integral and non-uniruled generic fiber, then $K_{X/T}$ is pseudo-effective.
The proof is based on covering $X$ with rational curves, which gives a contradiction as soon as both the base and the generic fiber are not uniruled. However, we assume only that the generic fiber is not uniruled. Hence, the hardest part of the proof is to show that there is a finite smooth non-uniruled cover of the base for which we show the following: If $T$ is a smooth projective variety over $k$ and $\mathcal{A}$ is an ample enough line bundle, then a cyclic cover of degree $p \nmid d$ given by a general element of $\left|\mathcal{A}^d\right|$ is not uniruled. For this we show the following cohomological uniruledness condition, which might be of independent interest: A smooth projective variety $T$ of dimenion $n$ is not uniruled whenever the dimension of the semi-stable part of $H^n(T, \mathcal{O}_T)$ is greater than that of $H^{n-1}(T, \mathcal{O}_T)$.
Additionally, we also show singular versions of all the above statements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_07158 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic Patakfalvi, Zsolt Algebraic Geometry 14E99, 14G17, 14J40, 14F99 We show that if $f\colon X \to T$ is a surjective morphism between smooth projective varieties over an algebraically closed field $k$ of characteristic $p>0$ with geometrically integral and non-uniruled generic fiber, then $K_{X/T}$ is pseudo-effective. The proof is based on covering $X$ with rational curves, which gives a contradiction as soon as both the base and the generic fiber are not uniruled. However, we assume only that the generic fiber is not uniruled. Hence, the hardest part of the proof is to show that there is a finite smooth non-uniruled cover of the base for which we show the following: If $T$ is a smooth projective variety over $k$ and $\mathcal{A}$ is an ample enough line bundle, then a cyclic cover of degree $p \nmid d$ given by a general element of $\left|\mathcal{A}^d\right|$ is not uniruled. For this we show the following cohomological uniruledness condition, which might be of independent interest: A smooth projective variety $T$ of dimenion $n$ is not uniruled whenever the dimension of the semi-stable part of $H^n(T, \mathcal{O}_T)$ is greater than that of $H^{n-1}(T, \mathcal{O}_T)$. Additionally, we also show singular versions of all the above statements. |
| title | Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic |
| topic | Algebraic Geometry 14E99, 14G17, 14J40, 14F99 |
| url | https://arxiv.org/abs/2009.07158 |