8 problems

Let $C$ be a curve of type $(a,b)$ in $\mathbb{P}^1\times\mathbb{P}^1$, and let $H_{(1,0)\cup(0,1)}$ be the curve of type $(2(a-1),2(b-1))$ whose intersection with $C$ has total mu…

Modified-weight characterization. Whenever $g \gg \ka$, $\sss$ is $\ka$-hyperelliptic if and only if

Atkin's conjecture. The cusp $\infty$ is a Weierstrass point of $X_0^+(p)$ for every prime $p>389$.

Let $S$ be a numerical semigroup of genus $g$, and let $\mathcal{M}^S_{g,1}$ be the corresponding locus in $\mathcal{M}_{g,1}$. The $2g$ codimension conjecture. If…

Let $S$ be a numerical semigroup of genus $g$, and let $\mathcal{M}^S_{g,1}$ denote the locus in $\mathcal{M}_{g,1}$ of pointed smooth curves whose Weierstrass semigroup contains,…

Let $(a^b)$ denote the rectangular partition with $b$ parts each equal to $a$, and let $\delta$ denote the difficulty invariant for partitions. Bounded difficulty conjecture. There…

Uniform codimension conjecture. The codimension of every component is bounded linearly in $g$ and by the partition size, with equality in the small-partition range. This conjecture…

Let ${\cal M}_g$ be the moduli space of smooth genus $g$ algebraic curves. For each $n$, let ${\cal W}_n\subset {\cal M}_g$ be the locus of curves having a Weierstrass point of ord…