7 problems

Let $Y$ be a nodal curve with $b4$ nodes, let $\Gamma$ be its dual graph, and let $\mathcal B$ be the set of bonds of $\Gamma$. Define the vector-weight collection…

Let $B$ be a family of curves satisfying the hypotheses ${\bf DIM}_{\mathrm{KP}}$: the locus of non-reduced curves has codimension greater than $\delta$, and for every equisingular…

Let $X$ be a K3 surface, let $L$ be a primitive polarization of genus $g$, and let $\mathcal{V}^n_{g,k}$ denote the moduli space of stable maps from smooth curves to $X$ that are u…

Bounded-nodes existence conjecture. Fix integers $r\geq 3$ and $t\geq 0$. There is a function $\mathbb{O}_{r,t}:\mathbb{N}\to\mathbb{N}$ such that…

Asymptotic existence conjecture. Fix an integer $r\geq 3$. There is a function $\mathbb{O}_r:\mathbb{N}\to\mathbb{N}$ such that $\lim_{g\to+\infty}\mathbb{O}_r(g)/g=0$ and an integ…

For each $n\geq1$, write … for integers $D_n,E_n,F_n,G_n$, and define the sequences $D=\\{D_{n+1}/D_n\\}_{n\geq1}$, $E=\\{E_{n+1}/E_n\\}_{n\geq1}$, $F=\\{F_{n+1}/F_n\\}_{n\geq1}$,…

Let $S$ be a smooth, irreducible, complex projective surface, let $4\mathscr{L}4$ be a line bundle on $S$, and write $Y=|\mathscr{L}|=\mathbb{P}(H^0(S,\mathscr{L}))$ with…