Interpolation for restricted tangent bundles of general curves

Larson, Eric

doi:10.2140/ant.2016.10.931

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Interpolation for restricted tangent bundles of general curves

Eric Larson

Vol. 10 (2016), No. 4, 931–938

DOI: 10.2140/ant.2016.10.931

Abstract

Let $q_{1}, q_{2}, \dots, q_{n} \in ℙ^{r}$ be a general collection of points, and $(C, p_{1}, p_{2}, \dots, p_{n})$ a general marked curve of genus $g$ . We determine when there exists a nondegenerate degree- $d$ map $f : C \to ℙ^{r}$ such that $f (p_{i}) = q_{i}$ for all $i$ . This is a consequence of our main theorem, which states that the restricted tangent bundle $f^{*} T_{ℙ^{r}}$ of a general curve of genus $g$ , equipped with a general degree- $d$ map $f$ to $ℙ^{r}$ , satisfies the property of interpolation, i.e., that for a general effective divisor $D$ of any degree on $C$ , either $H^{0} (f^{*} T_{ℙ^{r}} (- D)) = 0$ or $H^{1} (f^{*} T_{ℙ^{r}} (- D)) = 0$ . We also prove an analogous theorem for the twist $f^{*} T_{ℙ^{r}} (- 1)$ .

Keywords

restricted tangent bundle, interpolation

Mathematical Subject Classification 2010

Primary: 14H99

Milestones

Received: 30 November 2015

Accepted: 16 May 2016

Published: 20 June 2016

Authors

Eric Larson
Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139-4307 United States

Vol. 10, No. 4, 2016