$K3$ curves with index $k>1$

TL;DR

The paper investigates the forgetful maps $c_g^k:\mathcal{KC}_g^k\to\mathcal{M}_g$ for $k>1$, where $\mathcal{KC}_g^k$ classifies genus $g$ curves on $K3$ surfaces with index $k$. It derives explicit fibre-dimension formulas for the general fibre, distinguishing when $C$ lies on complete intersections or Mukai varieties; it also connects these to universal extensions and to $k$-spin (theta-characteristic) data, asking whether the image coincides with the locus of $k$-spin curves with many sections. The complete-intersection cases ($g_1=3,4,5$) yield concrete fibre dimensions and universal extensions, while the Mukai-model cases ($g_1=6,\,7,\,8,\,9,\,10$) reveal birationality phenomena and a conjectural finite-model behavior under automorphisms. The spin-curve analysis shows that, for $k=2$, the image has codimension $\binom{g_1+1}{2}$ in $\mathcal{M}_g$, with many components arising from special loci (hyperelliptic, bielliptic, double covers, etc.), and the paper provides partial classifications for $g_1=3,4$ and partial results for $k=3$, uncovering rich geometric structures on rational normal scrolls and quadrics. Overall, the work clarifies when K3 curves of index $k$ can be extended to higher-dimensional ambient varieties and how these extensions interplay with spin-structures, with implications for moduli and birational geometry of the relevant parameter spaces.

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.