On the unirationality of moduli spaces of pointed curves

Hanieh Keneshlou,Fabio Tanturri

doi:10.1007/s00209-021-02741-7

Hanieh Keneshlou, Fabio Tanturri

Open Access

https://doi.org/10.1007/s00209-021-02741-7

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Abstract

We show that mathcal {M}_{g,n}, the moduli space of smooth curves of genus g together with n marked points, is unirational for g=12 and 2 le nle 4 and for g=13 and 1 le n le 3, by constructing suitable dominant families of projective curves in mathbb {P}^1 times mathbb {P}^2 and mathbb {P}^3 respectively. We also exhibit several new unirationality results for moduli spaces of smooth curves of genus g together with n unordered points, establishing their unirationality for g=11, n=7 and g=12, n =5,6.

Highlights

The geometry of algebraic curves varying in families is a very fascinating and old topic, dating back to the nineteenth century
The interest around this subject naturally led to the definition of the moduli space Mg of smooth curves of genus g over the complex numbers
The study of its birational geometry has become a very active research area, especially after the unexpected results of Harris– Mumford–Eisenbud [15,24]: they showed that Mg is of general type for g ≥ 24, contradicting a long-standing conjecture by Severi about its unirationality for any g

Summary

Introduction

The geometry of algebraic curves varying in families is a very fascinating and old topic, dating back to the nineteenth century. It is natural to try to characterise the birational geometry and to determine the Kodaira dimension of the finitely many remaining cases for each g This problem has been investigated from many point of views and several contributions have been provided; we postpone to Sect. The family of curves of genus 13 is obtained building upon the unirationality of C10,6, granted by [3] and for which we exhibit an alternative proof in the second part of the paper. For a suitable choice of m < n, we exhibit a dominant unirational family of curves of genus g and degree 2g − 2 − m in Pg−m−1, reproving the unirationality of Mug,m; we 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 Mug,n for m < n ≤ m + m.

Brill–Noether Theory

Liaison

Computational verifications

Geiß’ construction