On Fujita invariants of subvarieties of a uniruled variety

TL;DR

This paper proves that on a smooth uniruled projective variety with a big semiample divisor, subvarieties with larger Fujita invariants are contained in a proper closed subset, revealing a boundedness property.

Contribution

It establishes a boundedness result for subvarieties with larger Fujita invariants in uniruled varieties, extending understanding of their geometric structure.

Findings

Subvarieties with higher Fujita invariants are contained in a proper closed subset.

The result applies to smooth uniruled projective varieties with big semiample divisors.

Provides a new perspective on the distribution of subvarieties based on Fujita invariants.

Abstract

We show that if $X$ is a smooth uniruled projective variety and $L$ a big and semiample $\mathbb{Q}$-divisor on $X$, then there exists a proper closed subset $W\subset X$ such that every subvariety $Y$ satisfying $a(Y,L)> a(X,L)$ is contained in $W$.