17 problems

Let $X$ be a surface, let $\mathcal{L}$ be a line bundle on $X$, and let $\delta$ be a non-negative integer. Let $N_\delta^X(\mathcal{L})$ be the number of curves in…

For a smooth projective plane, let $N^{d,\delta}$ denote the Severi degree, namely the number of possibly reduced $\delta$-nodal degree-$d$ curves passing through general…

Let $Y$ be an irreducible projective nodal curve with nodes $y_j$, let $I_j$ be the ideal sheaf of $y_j$, and let $U$ be the moduli space of torsion-free sheaves of rank $r$ and de…

Let $\Sigma_e$ be a rational ruled surface, let $E$ be the curve with $E^2=-e$, and let $F$ be a fiber of the ruling. Set $t^e_\delta(n,m):=t^{\Sigma_e}_\delta(\mathcal O(nF+mE))$,…

Let $\Sigma_e$ be a rational ruled surface, let $E$ be the curve with $E^2=-e$, and let $F$ be a fiber of the ruling. Set $t^e_\delta(n,m):=t^{\Sigma_e}_\delta(\mathcal O(nF+mE))$.…

Let $N^{d,\delta}$ be the number of plane curves of degree $d$ with $\delta$ nodes passing through $(d^2+3d)/2-\delta$ general points, including reducible curves, and write…

Good-polarization conjecture. The property that the structure sheaf $\mathcal{O}_C$ is $\operatorname{\underline{w}}$-stable should hold for any nodal curve $C$.

Kleiman–Piene conjecture. For every $r$, the class $[U(r)]$ is given by a universal polynomial in the classes $y(a,b,c)$.

Let $C$ be a nodal curve and let ${\underline w}$ be a polarization on $C$. A polarization is good if $\Delta_{\underline w}(E)\geq 0$ for every depth one sheaf $E$ on $C$, with eq…

Stability-goodness conjecture. The polarization $\underline w$ is good if and only if $\mathcal{O}_C$ is $\underline w$-stable.

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…

Let $f:X\to Y$ be the composition of the embedding $\iota:X\hookrightarrow F$ and the projection $F=S\times Y\to Y$. Let $m_r$ be the $r$-point cycle class of $f$, and let…

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…