23 problems

Global generation conjecture. For any $r\geq 1$, $\mathcal{O}_{U}(k\Theta_N)$ is globally generated on $U_X(r,0)$ for $k\geq r+1$.

Let $(P,\Xi)$ be a Prym variety associated with a double cover $\pi:\widetilde C\to C$, and let $\Sigma$ be the relevant Prym-theta subvariety. Let $\Gamma_{00}'$ consist of sectio…

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 $\Gamma_{00}\subset\Gamma$…

Casalaina-Martin's conjecture. If $(A,\Theta)$ is an indecomposable principally polarized abelian variety, then

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

Let $C$ be a reducible curve of arithmetic genus $g$, and let the associated theta divisor be constructed as in the paper. Consider an irreducible component of this theta divisor.…

Debarre's conjecture. If

Pareschi–Popa conjecture. If $\Theta=X+Y$ decomposes nontrivially as a sum, then either $(A,\Theta)$ is the Jacobian of a smooth projective curve, or $(A,\Theta)$ is the intermedia…

Let $L(g)$ be the Kronecker product of $g$ copies of the matrix $M^+(1)$, and let $h_0$ be the quantity defined by … Here $B$ is the matrix appearing in the preceding proposition.…

Salvati Manni–Grushevsky codimension conjecture. The loci $F_{3,0}^{(g)}$ and $I^{(g)}$ are purely of codimension $g$ in ${\mathcal A}_g$.

Folk conjecture on triple points.

Let $(X,\Theta)$ be a principally polarized abelian variety over $\mathbb C$ of dimension $g\geq 2$ with smooth symmetric theta divisor $\Theta$. For the variations…

Let $(X,\Theta)$ be a principally polarized abelian variety over $\mathbb C$ of dimension $g\geq 2$ with smooth symmetric theta divisor $\Theta$. For $x\in X(\mathbb C)$, set … whe…

Let $\mathcal A_g$ be the moduli space of principally polarized abelian varieties of genus $g$, and let $I^{(g)}\subset\mathcal A_g$ be the locus of pairs $(A,\Theta)$ whose theta…

Theta-multiplicity conjecture.

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

Let $(J(T),\Xi)$ be the intermediate Jacobian of a general Verra threefold $T$, let $\Xi$ be its theta divisor, and let $e$ be the canonical extension class; write $\Xi_e$ for the…

Let ${\mathcal A}_g$ be the moduli space of principally polarized abelian varieties, and let $G_k$ denote the loci introduced in the paper in analogy with the Andreotti–Mayer loci.…

Let ${\mathcal X}_g$ be the universal principally polarized abelian variety, and let $\Theta_{\rm sing}\subseteq {\mathcal X}_g$ be the locus of points lying on a theta divisor tog…

Let ${\mathcal A}_g$ be the moduli space of principally polarized abelian varieties, let ${\mathcal X}_g$ be the universal principally polarized abelian variety, and let…

Let $X$ be a smooth projective curve, and let $SU_X(r,d)$ denote the moduli space of semistable vector bundles of rank $r$ and degree $d$, with generalized theta line bundle…

Theta-multiplicity conjecture. The maximal multiplicity of the theta function for indecomposable ppavs is