# Fortnightly links (160)

Chandranandan Gangopadhyay, Ronnie Sebastian: Picard groups of some Quot schemes shows that the Picard group of the Quot scheme $\operatorname{Quot}_C(\mathcal{E},k,d)$ where $\mathcal{E}$ is a vector bundle of rank $r$ and degree $e$ on a smooth projective curve $C$ parametrising quotients of rank $k$ and degree $d$ is isomorphic to $\operatorname{Pic}(\operatorname{Jac}C)\times\mathbb{Z}\times\mathbb{Z}$, provided that $d\gg0$, $k\leq r-2$ and either $k\geq 2$ and $g\geq 3$ or $k\geq 3$ and $g\geq 2$.

It is also possible to fix the determinant of the quotient, so that one gets a moduli space somewhat similar to the moduli space of (semi)stable vector bundles on a curve with fixed determinant. And indeed, this brings me to a question, given that the Picard group then becomes isomorphic to $\mathbb{Z}\times\mathbb{Z}$.

**Question**Is this moduli space ever a smooth projective Fano variety?If it were rank 1, it would be either Fano or anti-Fano. As it is rank 2, it is not automatically (anti-)Fano. The methods are similar to how these things were studied for the moduli space of vector bundles by Drezet–Narasimhan, so I would hope that the answer is within reach. Let me know your thoughts!

Severin Barmeier, Zhengfang Wang: $\mathrm{A}_\infty$ deformations of extended Khovanov arc algebras and Stroppel's conjecture computes the Hochschild cohomology of an important class of associative algebras that appears in various fields, through a rather impressive tour de force. The paper ends with interesting speculation in Section 5.3, on deformation theory in symplectic geometry. Cool!