Narrow Results

The goal of this work is to study geometric properties of geometrically irreducible subschemes on degenerations of Fano varieties (more generally, of separably rationally connected varieties). It is known that these geometrically irreducible subschemes exist when the ground field has characteristic zero or contains an algebraically closed subfield. We show that the dimension of this geometrically irreducible subscheme has a lower bound by the Fano index of the generic fiber.

Let $X$ be a curve defined over $\mathbb{Q}$ and let $t\in \mathbb{Q}(X)$ be a non-constant rational function on $X$ of degree $v\ge 2$. For every rational number $a/b$ pick a point $P_{a/b}\in X(\overline{\mathbb{Q}})$ such that $t(P_{a/b})=a/b$. In this paper, we obtain lower bounds on the number of distinct fields among $\mathbb{Q}(P_{a/b})$ with $1\le a,b\le N$ under some assumptions on $t$. We show that if $t$ has a pole of order at least 2 or if there is a rational number $\alpha$ such that $t-\alpha$ has a zero of order at least 2, then the set $\{\mathbb{Q}(P_{a/b})|1\le a,b\le N\}$ contains $\gg \frac{N^2}{{(logN)}^2}$ elements. We also obtain partial results when $t$ does not have a pole of order at least two.