Relatively Very Free Curves and Rational Simple Connectedness

TL;DR

This paper investigates conditions under which relatively free rational curves imply the existence of very free ones, leading to results on rational simple connectedness for certain algebraic varieties, with applications to families over surfaces.

Contribution

It establishes that relatively free rational curves imply very free ones under specific conditions, and applies this to prove rational simple connectedness for certain complete intersections.

Findings

Relatively free curves imply very free curves when fibers have Picard number 1.

Smooth complete intersections with sum of squares of degrees ≤ n are strongly rationally simply connected.

Families over surfaces with vanishing Brauer obstruction have rational sections.

Abstract

Given a morphism between smooth projective varieties $f: W \to X$, we study whether $f$-relatively free rational curves imply the existence of $f$-relatively very free rational curves. The answer is shown to be positive when the fibers of the map $f$ have Picard number 1 and a further smoothness assumption is imposed. The main application is when $X \subset \PP^n$ is a smooth complete intersection of type $(d_1, ..., d_c)$ and $\sum d_i^2 \leq n$. In this case, we take $W$ to be the space of pointed lines contained in $X$ and the positive answer to the question implies that $X$ contains very twisting ruled surfaces and is strongly rationally simply connected. If the fibers of a smooth family of varieties over a 2-dimensional base satisfy these conditions and the Brauer obstruction vanishes, then the family has a rational section (see \cite{dJHS})