# On the rigidity of moduli of weighted pointed stable curves

## TL;DR

This paper investigates the rigidity of Hassett moduli spaces of weighted stable curves, showing that certain spaces are rigid over any field or characteristic zero, and analyzing deformations of special degenerations like the Segre cubic.

## Contribution

It establishes rigidity results for Hassett moduli spaces of weighted pointed stable curves and studies deformations of degenerations including the Segre cubic hypersurface.

## Key findings

- _{g,A[n]} is rigid over any field for g .
- _{g,A[n]} is rigid over characteristic zero for g  1.
- The Segre cubic has a 10-dimensional smooth deformation space.

## Abstract

Let $\overline{\mathcal{M}}_{g,A[n]}$ be the Hassett moduli stack of weighted stable curves, and let $\overline{M}_{g,A[n]}$ be its coarse moduli space. These are compactifications of $\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, obtained by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \leq 1$ to the markings; they are defined over $\mathbb{Z}$, and therefore over any field. We study the first order infinitesimal deformations of $\overline{\mathcal{M}}_{g,A[n]}$ and $\overline{M}_{g,A[n]}$. In particular, we show that $\overline{M}_{0,A[n]}$ is rigid over any field, if $g\geq 1$ then $\overline{\mathcal{M}}_{g,A[n]}$ is rigid over any field of characteristic zero, and if $g+n > 4$ then the coarse moduli space $\overline{M}_{g,A[n]}$ is rigid over an algebraically closed field of characteristic zero. Finally, we take into account a degeneration of Hassett spaces parametrizing rational curves obtained by allowing the weights to have sum equal to two. In particular, we consider such a Hassett $3$-fold which is isomorphic to the Segre cubic hypersurface in $\mathbb{P}^4$, and we prove that its family of first order infinitesimal deformations is non-singular of dimension ten, and the general deformation is smooth.