K3 curves with index k>1

Ciro Ciliberto,Thomas Dedieu

doi:10.1007/s40574-021-00286-5

Abstract

Let mathcal {KC}_g ^k be the moduli stack of pairs (S, C) with S a K3 surface and Csubseteq 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.

Highlights

We provide a description of the universal extension X as a complete intersection in a weighted projective space, embedded by a suitable divisor in Pic(X ) of the anticanonical class
We show that in cases (a) k = 1 and g1 > 6, and (b) k = 2 and g1 = 6, the Mukai variety is the universal extension of the general membre of im(cgk ); this was certainly a natural thing to guess, but as far as we know this had not been proved yet
We show that the Mukai variety in P(Ug1 ) is the universal extension of a general canonical curve of genus g1

Summary

Introduction

3) For g1 ∈ {3, 4, 5} the general membre C of the image of cgk , with g = 1 + (g1 − 1)k2, has a model as a complete intersection in Pg1 This allows for a direct description of the fibre (ggk )−1(C); in particular we can compute its dimension, which gives cork( C ), recovering the values obtained in [9]. We provide a description of the universal extension X as a complete intersection in a weighted projective space, embedded by a suitable divisor in Pic(X ) of the anticanonical class This works in the exceptional cases of (1.1.1), in which cases we obtain interesting examples showing the sharpness of Lvovski’s main theorem in [20]. See [17, Cor. 15.2.12] for (2.4.2), and [17, Chap. 5 Prop. 3.3] for (2.4.1)

Useful results

Curves on Mukai varieties

SL6 35

Results and proofs

5.10 Theorem