[Paper Review] $K3$ curves with index $k>1$.

Ciro Ciliberto, Thomas Dedieu|arXiv (Cornell University)|Dec 19, 2020

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper studies K3 curves of index $k > 1$ by analyzing the forgetful map $c^k_g: \mathcal{KC}^k_g \to \mathcal{M}_g$, computing the dimension of its general fiber geometrically. The key result is that the fiber dimension is non-trivial only when the K3 surface is a complete intersection or a section of a Mukai variety, leading to classification results for $k$-spin curves and explicit constructions of universal extensions as complete intersections in weighted projective 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.

Motivation & Objective

• To compute the dimension of the general fiber of the forgetful map $c^k_g: \mathcal{KC}^k_g \to \mathcal{M}_g$ for $k > 1$, extending prior work on the primitive case ($k=1$).
• To investigate whether $c^k_g$ dominates the locus of $k$-spin curves with $g_1 + 1$ independent sections, particularly in the case where the K3 surface is a complete intersection.
• To construct and describe the universal extension $X$ of a general $K3$ curve of index $k > 1$, especially in cases where $S$ is a complete intersection or a section of a Mukai variety.
• To provide geometric constructions of universal extensions as complete intersections in weighted projective spaces, recovering and refining results from cohomological computations in [9].
• To explore modular properties of Mukai varieties and their sections, particularly in relation to the automorphism group $G = \mathrm{Aut}(M_{g_1})$ and the conjugacy of models of $C$.

Proposed method

• Geometric analysis of the Gauss–Wahl map $\Phi_C$ to compute the corank, which determines the fiber dimension of $c^k_g$.
• Explicit construction of the universal extension $X$ as a complete intersection in a weighted projective space, using the anticanonical divisor class.
• Classification of the general fiber $ (c^k_g)^{-1}(C) $ by analyzing the linear systems on $S$ and their pullbacks to $C$, particularly in the case where $S$ is a complete intersection.
• Use of Prokhorov’s bound on the genus of Gorenstein Fano threefolds to restrict the range of $g$ for which $c^k_g$ can have positive-dimensional fibers.
• Factorization of $c^k_g$ through the moduli stack $S^{1/k}_{g_1,g}$ of $k$-spin curves with $h^0(\theta) \geq g_1 + 1$, enabling comparison with known moduli of spin curves.
• Case-by-case analysis of surface models (e.g., rational scrolls, del Pezzo surfaces, blow-ups of $\mathbb{P}^2$) to rule out configurations leading to $g^1_8$ and thus constrain fiber structure.

Experimental results

Research questions

• RQ1 For which values of $g_1$ and $k > 1$ is the general fiber of $c^k_g: \mathcal{KC}^k_g \to \mathcal{M}_g$ non-trivial in dimension?
• RQ2 Can the universal extension $X$ of a general $K3$ curve $C$ of index $k > 1$ be explicitly constructed as a complete intersection in a weighted projective space?
• RQ3 Does the Mukai variety $M_{g_1}$ serve as the universal extension of the general curve in the image of $c^k_g$ for $k=1$, $g_1 > 6$, and $k=2$, $g_1=6$?
• RQ4 Are all models of a general $K3$ curve $C$ of index $k > 1$ conjugate under the action of $\mathrm{Aut}(M_{g_1})$ when $C$ is a section of a Mukai variety?
• RQ5 To what extent does the forgetful map $c^k_g$ dominate the moduli space of $k$-spin curves with $g_1 + 1$ independent sections?

Key findings

• The general fiber of $c^k_g$ has positive dimension only when $g_1 \in \{3,4,5\}$ (complete intersection case) or $g_1 \in \{6,7,8,9,10\}$ (Mukai variety case), with $g > 37$ implying generic injectivity except for finitely many $g_1, k$ pairs.
• For $g_1 \in \{3,4,5\}$, the universal extension $X$ of a general $K3$ curve $C$ is a complete intersection in a weighted projective space, embedded via a divisor in $| -K_X |$, and this construction recovers the corank of the Gauss–Wahl map computed cohomologically in [9].
• For $g_1 = 6$, $k = 2$, the general fiber of $c^k_g$ has dimension 1, and the Mukai variety $M_6$ is the universal extension of the general curve in the image.
• In the case $g_1 = 6$, $k = 2$, the moduli stack $S^{1/2}_{6,g}$ is dominated by $\mathcal{KC}^2_g$, and the general curve in the image is a $k$-spin curve with $h^0(\theta) = 7$.
• The Mukai variety $M_{g_1}$ is the universal extension of the general curve in $\mathrm{im}(c^k_g)$ when $k=1$, $g_1 > 6$, and when $k=2$, $g_1 = 6$, confirming a natural conjecture.
• Explicit examples of $k$-spin curves in $S^{1/k}_{g_1,g}$ with $h^0(\theta) = g_1 + 1$ are constructed, including cases with base points and singular models, showing the image of $c^k_g$ is strictly larger than the locus of base-point-free $k$-spin curves.