54 problems

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…

Debarre's conjecture. If

The general Hodge conjecture. The Hodge structure $\mathbb K_{\mathbb Q}=\mathbb K\otimes\mathbb Q$ is contained in the image, via Gysin pushforward, of the cohomology of a smooth,…

Let $(A,\Theta)$ be an irreducible principally polarized abelian variety of dimension $g$, and let $x\in\Theta$. Write $\operatorname{mult}_x(\Theta)$ for the multiplicity of…

Let $F$ be a number field that is Galois over $\bf Q$ or over an imaginary quadratic number field. Let $h^0$ be the arithmetic section-counting function on ${\rm Pic}^0(F)$, with…

For a smooth canonical curve, its Jacobian has a theta divisor whose double points have tangent cones, viewed as quadrics in the canonical space, and the canonical curve has an ide…

Serre's conjecture. The difference of the motives of $\Theta$ and $\Theta'$ should be divisible by the Lefschetz motive:

Let $(A,\Theta)$ be a principally polarized abelian variety over $\mathbb C$ of dimension $g\geq 4$ with smooth theta divisor $\Theta$. Let $K(\Theta,\mathbb Q)$ be the primitive p…

Let $\pi \colon \widetilde{C}\to C$ be a connected étale double cover of the smooth curve $C$, and let $L$ be a line bundle corresponding to the point $x\in P(\widetilde{C},\pi)$ w…

Let $C$ be the canonical curve and let $|F_4|$ denote the linear subsystem spanned by the quartic tangent cones $F_W$. The base locus of $|F_4|$ is the canonical curve $C$. Base-lo…

Popa's conjecture. The linear series $|k\Theta_N|$ should be base point free for $k\geq r+1$.

Let $X$ be a hyperelliptic curve of genus $g$, let $J(X)$ be its Jacobian variety, and let $\Theta$ be the theta divisor in $J(X)$. For $k\leq g$, write…

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 $\Gamma_{00}\subset\Gamma$…

Let $C$ be a smooth projective curve of genus $g$, let $J_C^d$ be its degree-$d$ Jacobian, and let $x\in J_C^d$ with $1\leq d\leq g$. Let ${\cal L}_x$ be the line bundle on $S^dC$…

Let $C$ be a smooth projective nonhyperelliptic curve of genus $4$ without vanishing theta-nulls. Let ${\rm SU}_C(2)$ be the moduli variety of semistable rank-$2$ vector bundles on…

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 $A$ be an abelian variety and let $X\subset A$ be an ample hypersurface. Consider the canonical map associated with the polarization given by $X$, and let the Pfaffian of this…

Let $(A,\Theta)$ be an indecomposable principally polarized abelian variety of dimension $g$, and let $x\in\Theta$. Grushevsky's multiplicity conjecture. The multiplicity of…

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.…

Let $\Delta$ satisfy the condition referred to as, let $F$ be integrally tempered, let $\bm{\mathcal{M}}$ be the relevant moduli space, let $\bm{\nu}(\bm{\mathcal{M}})$ be the imag…