15 problems

Ciliberto–van der Geer's conjecture. If $1\le \ell\le g-3$, then

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

Let ${\mathcal A}_4$ be the moduli space of 4-dimensional principally polarized abelian varieties, let $\tau\in{\mathcal A}_4$, and suppose that some theta constant at $\tau$ and i…

Let $(X,\Theta)$ be an irreducible principally polarized abelian variety, let $Y\subset X$ be a non-degenerate subvariety of dimension $m$, and write $\theta=c_1(\Theta)$. Say that…

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

Let $(A,\a0\Theta)$ be a principally polarized abelian variety of dimension $g\geq 2$, meaning that $\Theta$ is an ample divisor with $h^0(A,\mathcal{O}_A(\Theta))=1$. Write…

Let $(A,\Theta)$ be a principally polarized abelian variety, and for a fixed $p\in\mathbb P^{g-1}$ let $Z_p\subset |2\Theta|$ be the linear system generated by the divisors … where…

Pareschi--Popa's generic vanishing conjecture. The only geometrically nondegenerate GV-subschemes are the known examples of minimal class: the Brill--Noether loci $W_d(C)$ in Jacob…

Superfluity conjecture. The condition

Boundary-component conjecture. The locus

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.

$\Gamma_{00}$ conjecture. If