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

Let X be a very general hypersurface of degree d in Pn. We investigate positivity properties of the spaces Re(X) of degree e rational curves in X. We show that for small e, Re(X) has no rational curves meeting the locus of smooth embedded curves. We show that for n d, there are no rational curves other than lines in the locus Y X swept out by lines. We exhibit differential forms on a smooth compactification of Re(X) for every e and n 2 d 1 2(n + 1).

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hypersurface, rational curve, rational surface, birational geometry
Mathematical Subject Classification 2010
Primary: 14E08
Received: 30 April 2019
Revised: 14 August 2019
Accepted: 16 September 2019
Published: 29 May 2020
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