#### Vol. 10, No. 4, 2016

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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 $\left(C,{p}_{1},{p}_{2},\dots ,{p}_{n}\right)$ a general marked curve of genus $g$. We determine when there exists a nondegenerate degree-$d$ map $f:C\to {ℙ}^{r}$ such that $f\left({p}_{i}\right)={q}_{i}$ for all $i$. This is a consequence of our main theorem, which states that the restricted tangent bundle ${f}^{\ast }{T}_{{ℙ}^{r}}$ 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 ${H}^{0}\left({f}^{\ast }{T}_{{ℙ}^{r}}\left(-D\right)\right)=0$ or ${H}^{1}\left({f}^{\ast }{T}_{{ℙ}^{r}}\left(-D\right)\right)=0$. We also prove an analogous theorem for the twist ${f}^{\ast }{T}_{{ℙ}^{r}}\left(-1\right)$.

##### Keywords
restricted tangent bundle, interpolation
Primary: 14H99