21 problems

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…

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 $(P,\Xi)$ 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…

Prym theta-function isomorphism conjecture. These homomorphisms should be isomorphisms in both the odd- and even-rank cases.

Let $g\geq 7$, let ${\cal R}_g$ be the moduli space of étale double covers of smooth curves of genus $g$, and let ${\cal P}:{\cal R}_g\to{\cal A}_{g-1}$ be the Prym map. The tetrag…

Let $A\in{\cal A}_4$ be a principally polarized abelian variety, let $\Gamma_{00}$ be the van Geemen–van der Geer linear system on $A$, and let $X\subset\Gamma_{00}$ be the cubic t…

Let $X$ be a principally polarized Abelian variety with theta function $\theta$. Let $\widehat{\theta}_1$ denote the Prym theta function associated with the first involution, let…

Let $R_{\widetilde{G}/G}$ be the cone associated with a double cover $\widetilde{G}\to G$. As a double cover of tropical curves $\widetilde{\Gamma}\to\Gamma$ varies over…

Let $\widetilde{X}\to X$ be a double cover of algebraic curves or metric graphs, and let the Abel–Prym map of degree parameter $d$ be the map from the $d$-fold symmetric product to…

Let $\widetilde{\Gamma}\to\Gamma$ be a possibly dilated double cover of tropical curves, let $g_0=\dim \operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$, and for $1\leq d\leq g_0$ le…

The tropical Poincaré–Prym formula. The cycle class of $\widetilde{Y}_d$ satisfies

Let $\mathcal{R}_6$ be the moduli space of Prym curves of genus $6$, let $\mathcal{A}_5$ be the moduli space of principally polarized abelian varieties of dimension $5$, and let…

Let $C$ be a curve of genus $2$ or $3$ over a number field $K$, with an unramified double cover … and associated Prym variety $\operatorname{Prym}(D/C)$ over $K$. Let…

Prymian integrable systems are integrable systems whose Lagrangian fibres are Prym varieties, arising from natural involutions on moduli spaces of sheaves on K3 surfaces. The Hitch…

Let $S'$ be a K3 surface that is a branched double cover of a degree-one del Pezzo surface $T'$, and let $D'$ be a genus-two curve on $T'$ covered by a genus-five curve $C'$ on…

Let $\pi:\widetilde\Gamma\to\Gamma$ be a topological double cover, and let $V(\Gamma,\pi)$ be its Prym variety with two connected components. A divisor has rank $r(D)$ in the tropi…

Prym tropicalization conjecture. The skeleton of the Prym variety associated with $f$ is the tropical Prym variety associated with $\phi$; equivalently, the Prym construction commu…

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…

Donagi's conjecture. The restricted Prym map

Let $p:C\overset{\pi}{\longrightarrow}C'\overset{g}{\longrightarrow}\mathbb{P}^1$ be a simply ramified covering of type $B_n$. Let $P(C,C')$ be its Prym variety, let $P(X,\delta)$…

Let $p:C\overset{\pi}{\longrightarrow}C'\overset{g}{\longrightarrow}Y$ be a covering of smooth projective curves with $\deg(\pi)=2$ and $\deg(g)=n$, and let $P(X,\delta)$ be the Pr…