12 problems

Let $X$ be a principally polarized complex abelian variety, and let its Kummer variety be the quotient of $X$ by the involution $x\mapsto -x$. A trisecant line is a line meeting th…

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 $\bm A_g(2,4)$ be the moduli space of principally polarized abelian varieties with theta structure, and let $\Phi:\bm A_g(2,4)\to{\mathbb P}^{2^g-1}$ be the map defined by thet…

Let $g>1$. Denote by $\boldsymbol A_g$ the coarse moduli space of principally polarized abelian varieties of dimension $g$, and by $\boldsymbol{\mathsf K}^{\operatorname{em}}_g$ th…

Numerical moduli space conjecture. The preceding proposition holds for all $n$: there exists an abelian surface $A$ such that $X$ is the fibre of the Albanese map of a moduli space…

Sturmfels' generation conjecture. The prime ideal of the universal Kummer variety in genus $3$ is generated by $891$ bihomogeneous polynomials in $(u,x)$: one Schottky polynomial o…

The Kummer-coordinate addition relations conjecture. Suppose that $(P,Q)\in A\times A$ is a good pair. Then:

Let $k$ be a field of characteristic not equal to $2,3,5$, and let $F(X,Z)\in k[X,Z]$ be a binary octic form with $\deg_X(F)=7$. Let $d(4)$ denote the dimension of the space of qua…

Consider the moduli space of polarised abelian varieties and the associated odd-dimensional Kummer varieties. Here, “most” refers to the generic or dense family arising from absolu…

Let $\mathcal{I}_3$ be the prime ideal of the universal Kummer variety in the variables $({\bf u},{\bf x})$. The previously constructed equations consist of the Satake polynomial,…

Let $\mathcal{K}_3^{\rm flex}\subset \mathbb{P}^6\times\mathbb{P}^7$ be the universal Kummer threefold, with the previously constructed $8$ polynomials of bidegree $(7,3)$ and the…