27 problems

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 $J_g$ 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,

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…

Novikov's conjecture. The function $u(x,y,t)$ satisfies the KP equation if and only if $\Omega$ is the period matrix of some compact Riemann surface $\Sigma$.

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 $A$ be an indecomposable principally polarized abelian variety, and let $\operatorname{Kum}(A)$ be its Kummer variety. A trisecant is a line meeting the Kummer variety in three…

Let $A$ be an indecomposable principally polarized abelian variety, and let $\operatorname{Kum}(A)$ denote its Kummer variety. A trisecant line is a line meeting the Kummer variety…

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 $\mathbb{B}$ be a Riemann matrix and let Fay's identity be the theta-functional identity associated with the corresponding principally polarized abelian variety. The Novikov co…

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…

For $g\ge 3$, let ${\mathcal I}_g^A$ be the ideal of relations among the theta constants of a general principally polarized abelian variety. Riemann quartic generation conjecture.…

Let $\tau\in\mathbb H_g$ be a period matrix, and let $\theta$ be the corresponding Riemann theta function. The Kadomtsev–Petviashvili equation is the differential equation … where…

Let $(A,\Theta)$ be a principally polarised complex abelian variety of dimension $g$, and let … be the Andreotti–Mayer loci. Define the Gauss loci by … Here $\mathcal G$ is the Gau…

Let $\mathcal{RA}_g$ be the relevant moduli space of principally polarized abelian varieties equipped with a semiperiod, let $\beta_g:\mathcal{RA}_g\to\mathbb{P}_{g-1}$ and…

Let $g\geq 5$. Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties, let $\mathcal{J}_g$ be the Jacobian locus, and let $\mathcal{S}_g$ be the locus c…

Folk conjecture on Andreotti–Mayer loci. 1. All irreducible components of $N_{g-4}$ except the locus of Jacobians ${\mathcal J}_g$ are contained in the theta-null divisor…

$\Gamma_{00}$ conjecture. If

Theta-multiplicity conjecture.