Geometry of the moduli space of n-pointed K3 surfaces of genus 11

Abstract

We prove that the moduli space of polarized K3 surfaces of genus 11 with n marked points is unirational when n⩽6 and uniruled when n⩽7. As a consequence we settle a long standing but not proved assertion about the unirationality of M11,n for n⩽6. We also prove that the moduli space of polarized K3 surfaces of genus 11 with n⩾9 marked points has non‐negative Kodaira dimension.

On the unirationality of moduli spaces of pointed curves

“…For a suitable choice of m < n, we exhibit a dominant unirational family of curves of genus g and degree 2g − 2 − m in P g−m−1 , thus reproving the unirationality of M u g,m ; we then show that by performing liaison forth and back we can impose a certain number m of additional points on these curves, yielding the unirationality of M u g,n for m < n ≤ m + m . With the same general approach, we also exhibit (Remark 4.4) a constructive alternative proof of the unirationality of M u g,n for g = 10 and n = 6, 7, already achieved in [3].…”

“…We remark that in [27] it was previously claimed that M 11,n is unirational for n ≤ 10. However, Barros in [3] noticed that the original argument contained a flaw and only proves the uniruledness of M 11,n in that range. Barros managed to show that M 11,n is unirational for n ≤ 6, and that it is not unirational for n = 9 or n = 10.…”

“…We use projective models in P 3 and P 1 × P 2 for M 13,n , M 12,n , respectively. The family of curves of genus 13 is obtained building upon the unirationality of C 10,6 , granted by [3] and for which we exhibit an alternative proof in the second part of the paper. The construction of the family of curves of genus 12 is based on a particular construction in [22], which we recall in details.…”

“…4 we provide a brief account on the known results about its birational geometry. For what concerns the unirationality, it is known for g ≤ 9; beside the results easily deducible from M g,n , only the cases g = 10, n = 6, 7 [3] are known.…”

“…The number η(g) (respectively, τ (g)) denotes the smallest number of points n for which the Kodaira dimension of M g,n is known to be non-negative (respectively, maximal). The table is based on contributions given by [3,4,6,9,10,25,27,34] for the unirationality; [1,5,10,18,19,27] for the uniruledness; [8,16,19,20,27,32] for η(g) and τ (g).…”