$K3$ curves with index $k>1$

TL;DR

This paper explores the geometry of K3 surfaces with curves of divisibility greater than one, analyzing the fibers of the forgetful map to the moduli space of curves and classifying related spin curves.

Contribution

It computes the fiber dimensions of the forgetful map for K3 surfaces with special embeddings and classifies certain spin curves, revealing new geometric and modular properties.

Findings

Fiber dimension is interesting mainly for complete intersections or Mukai sections.

Existence of Fano varieties extending curves in canonical embeddings.

Classification results for k-spin curves in specific cases.

Abstract

Let $\mathcal{KC}_g ^k$ be the moduli stack of pairs $(S,C)$ with $S$ a $K3$ surface and $C\subset S$ a genus $g$ curve with divisibility $k$ in $\mathrm{Pic}(S)$. In this article we study the forgetful map $c_g^k:(S,C) \mapsto C$ from $\mathcal{KC}_g ^k$ to $\mathcal{M}_g$ for $k>1$. First we compute by geometric means the dimension of its general fibre. This turns out to be interesting only when $S$ is a complete intersection or a section of a Mukai variety. In the former case we find the existence of interesting Fano varieties extending $C$ in its canonical embedding. In the latter case this is related to delicate modular properties of the Mukai varieties. Next we investigate whether $c_g^k$ dominates the locus in $\mathcal{M}_g$ of $k$-spin curves with the appropriate number of independent sections. We are able to do this only when $S$ is a complete intersection, and obtain in these cases some classification results for spin curves.