On noncommutative rigidity of the moduli stack of stable pointed curves

As the first non-trivial case of the Kapranov's geometric syzygy principle, Hacking proved that the Grothendieck-Knudsen moduli stack of stable $n$-pointed genus $g$-curves is rigid for any $(g, n)$. In this talk we show that the $2$-nd Hochschild cohomology of the moduli space is trivial when $g=0$ except the case $n=5$, which implies the rigidity of the abelian category of coherent sheaves on the moduli space. We will also explain what is known for the case $g>0$ and the remaining issues. This is a joint work in progress with Taro Sano.