6 problems

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 $r+1$…

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

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.

Farkas's codimension conjecture. For $r\geq 1$ and $g\geq {r+2\choose 2}$, there exists a component of $\mathcal{S}_g^r$ having codimension ${r+1\choose 2}$ inside $\mathcal{S}_g$.…