Positivity results for spaces of rational curves

Beheshti, Roya; Riedl, Eric

doi:10.2140/ant.2020.14.485

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Positivity results for spaces of rational curves

Roya Beheshti and Eric Riedl

Vol. 14 (2020), No. 2, 485–500

DOI: 10.2140/ant.2020.14.485

Abstract

Let $X$ be a very general hypersurface of degree $d$ in $P^{n}$ . We investigate positivity properties of the spaces $R_{e} (X)$ of degree $e$ rational curves in $X$ . We show that for small $e$ , $R_{e} (X)$ has no rational curves meeting the locus of smooth embedded curves. We show that for $n \leq d$ , there are no rational curves other than lines in the locus $Y \subset X$ swept out by lines. We exhibit differential forms on a smooth compactification of $R_{e} (X)$ for every $e$ and $n - 2 \geq d \geq \frac{1}{2} (n + 1)$ .

Keywords

hypersurface, rational curve, rational surface, birational geometry

Mathematical Subject Classification 2010

Primary: 14E08

Milestones

Received: 30 April 2019

Revised: 14 August 2019

Accepted: 16 September 2019

Published: 29 May 2020

Authors

Roya Beheshti
Mathematics and Statistics Washington University in St. Louis Saint Louis, MO United States

Eric Riedl
Department of Mathematics University of Notre Dame Notre Dame, IN United States

Vol. 14, No. 2, 2020