Vol. 14, No. 2, 2020

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Positivity results for spaces of rational curves

Roya Beheshti and Eric Riedl

Vol. 14 (2020), No. 2, 485–500
Abstract

Let $X$ be a very general hypersurface of degree $d$ in ${P}^{n}$. We investigate positivity properties of the spaces ${R}_{e}\left(X\right)$ of degree $e$ rational curves in $X$. We show that for small $e$, ${R}_{e}\left(X\right)$ has no rational curves meeting the locus of smooth embedded curves. We show that for $n\le d$, 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 ${R}_{e}\left(X\right)$ for every $e$ and $n-2\ge d\ge \frac{1}{2}\left(n+1\right)$.

Keywords
hypersurface, rational curve, rational surface, birational geometry
Primary: 14E08