10 problems

Parity conjecture. The unique invariant theta characteristic on $X(p)$ is always even.

Let ${\mathcal M}_g$ be the moduli space of smooth curves of genus $g$, and let … be the locus of curves possessing an $r$-dimensional theta-characteristic. Farkas's conjecture. Fo…

Let $\mathcal{S}_g$ denote the moduli space of smooth genus-$g$ spin curves, and let $\mathcal{S}_g^r$ denote the locus of spin curves whose theta-characteristic has at least $r+1$…

For an odd prime $p\geq 7$, let $X(p)$ be the modular curve of genus … with its $PSL(2,p)$-action of signature $(0;2,3,p)$. An invariant characteristic is the unique characteristic…

Dimension-seven conjecture.

Let $\Gamma$ be a tropical sextic curve of genus $4$ in $\mathbb{R}^3$. Two tritangent planes to $\Gamma$ are equivalent when they determine equivalent tangency divisors on…

BLMPR conjecture. Each of the seven odd theta characteristics of $\Gamma$ is the specialization of four effective theta characteristics of $X$, counted with multiplicity.

Let $g$ be a positive integer, let $\mathfrak{H}_g$ be the Siegel upper half-space, and let $\Gamma_g=\operatorname{Sp}_{2g}(\mathbb Z)$ and $\Gamma_g(1,2)\subset\Gamma_g$. A modul…

Farkas–Verra conjecture. This limit is the union of the curve

Let $S(F)$ be the plane quartic under consideration, let $Ar$ be its associated Pfaffian representation, and let ${\mathcal L}_{\vartheta}$ be an even theta characteristic on…