16 problems

Let $(B,\Xi)$ be a simple principally polarized abelian variety of dimension $g$, and suppose that $(B,\Xi)\notin N_{g,i}$ for every $i\geq k\geq1$. Here $N_k(B,\Xi)\subset B$ is t…

Let $g$ and $k$ be integers with $1\leq k\leq g-3$, and let $N$ be an irreducible component of the Andreotti–Mayer locus $N_{g,k}$ whose general point corresponds to a principally…

The Prym characterization by the minimal number of conditions. Then $n\geq 2g$, and equality characterizes Prym varieties. This is proposed as a higher Castelnuovo–Schottky problem…

The minimal-class characterization by strong theta-regularity. The subvariety $Y$ represents a minimal class if and only if its ideal sheaf is strongly $3$-$\Theta$-regular. This p…

Let $(P,\Xi)$ be a principally polarized abelian variety of dimension $p\geq 4$, and let $V_{\inf,00}$ be the base locus of the linear system of quartic tangent cones at the origin…

Let $(P,\Xi)$ be a principally polarized abelian variety of dimension $p\geq 4$, with $\Xi$ a symmetric theta divisor. Let $\Gamma=H^0(P,2\Xi)$, and let $\Gamma_{00}\subset\Gamma$…

Let a principally polarized indecomposable abelian variety be given, with its Kummer variety and associated theta functions. A family of flex lines has a fourth-order jet when its…

Let $\mathcal{J}_g$ denote the locus of $g$-dimensional Jacobians in the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties, and let…

Debarre's strong Schottky conjecture. If $(A,\Theta)$ is not a hyperelliptic Jacobian or the intermediate Jacobian of a smooth cubic threefold, then

Let $\mathcal{H}_\mathcal{C}^{M}$ be the main component of the Hirota variety $\mathcal{H}_\mathcal{C}$, and let $\mathcal{H}_\mathcal{C}^{I}$ denote its distinguished irreducible…

Let $g$ be a genus, and let $\mathcal{H}_\mathcal{C}$ be the Hirota variety associated with the rational nodal curve data, with main component $\mathcal{H}_\mathcal{C}^{M}$ and the…

Let $J_g$ be the Jacobian locus in $\mathcal A_g$, and let $\mathcal S_g$ be the small Schottky–Jung locus, defined as the intersection of the Schottky–Jung loci associated with al…

Van Geemen–van der Geer's conjecture. If $(A,\Theta)$ is the Jacobian of a smooth curve $C$, then, as a set,

$\Gamma_{00}$ conjecture. If

Theta-multiplicity conjecture.

Debarre's codimension conjecture. For any $1\leq k\leq g-3$,