Interpolation for Restricted Tangent Bundles of General Curves

TL;DR

This paper characterizes when a general curve can be mapped to projective space with specified points, showing that the restricted tangent bundle satisfies the interpolation property, which has implications for the existence of such maps.

Contribution

It proves that the restricted tangent bundle of a general curve with a general map satisfies interpolation, extending understanding of curve mappings in algebraic geometry.

Findings

The restricted tangent bundle satisfies interpolation for general curves and maps.

Conditions are established for the existence of maps with prescribed images of marked points.

An analogous result is proved for the twisted bundle f^* T_{P^r}(-1).

Abstract

Let (C, p_1, p_2, \ldots, p_n) be a general marked curve of genus g, and q_1, q_2, ..., q_n \in P^r be a general collection of points. We determine when there exists a nondegenerate degree d map f : C \to P^r so 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).