16 problems

Let $X=E_1\times E_2\times E_3$ be a product of three Tate elliptic curves with parameters $q_i$ for $i=1,2,3$. Let $r$ be the rank of the space of integer triples $(n_1,n_2,n_3)$…

Griffiths's conjecture on incidentally trivial cycles. For every $p\in\{1,\ldots,d\}$ and every $z\in I^p(X)$, there is a positive integer $N$ such that

Jacobi-type theorem. There is some positive integer $k$ such that the topological Abel--Jacobi mapping

Let $Y$ be a quartic double solid and let $X$ be a GM threefold. Let $J(Y)$ and $J(X)$ denote their intermediate Jacobians, each with its principal polarization. Intermediate-Jacob…

Let $X$ be the fourfold considered in the paper, let $J^2(X)$ be its intermediate Jacobian, and let ${\mathsf L}:J^2(X)\to J^3(X)$ be induced by intersection with an ample cycle. D…

Let $X$ and $X'$ be prime Fano threefolds of index one or two, with intermediate Jacobians $J(X)$ and $J(X')$ and Kuznetsov components $\mathcal{K}u(X)$ and $\mathcal{K}u(X')$. Cat…

Let $\mathcal{X}_{10}$ be the moduli space of ordinary Gushel–Mukai threefolds, let $\mathcal{A}_{10}$ be the moduli space of $10$-dimensional principally polarised abelian varieti…

Castravet's conjecture. The maximally rationally connected quotient of ${\mathcal C}$ is birationally equivalent to the twisted intermediate Jacobian of $Y$.

Let $X$ be a threefold satisfying the stated properties, including a decomposition of its intermediate Jacobian … where the $S_i$ are smooth projective curves of positive genus and…

Let $X$ be an $n$-dimensional connected scheme smooth and projective over ${\bf C}$, with $n\geq 2$. Write $CH_{Jac,\mathbf Q}^2(X)$ for the kernel of the Abel–Jacobi map on homolo…

Let $X$ be an $n$-dimensional connected scheme smooth and projective over ${\bf C}$. Let $M_2(X)^*$ and $i_{2n-2}:M_2(X)^*\otimes {\bf L}^{n-2}\to M(X)$ be as obtained from the dec…

Yui's conjecture. Then

Let $X$ and $Y$ be schemes as in Example (E8), and let $J(X)$ and $J(Y)$ denote their intermediate Jacobians. The E8 verepresentability and Torelli conjecture. The scheme $Y$ is ve…

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 $X_{10}$ be a general Fano threefold of degree $10$, and let a quartic double solid be a smooth double cover of $\mathbb{P}^3$ branched over a quartic surface. Tyurin's conject…

Debarre's conjecture. The cycle $Y$ has minimal cohomology class if and only if $(A,\Theta)$ is the polarised Jacobian of a curve $C$ of genus $n$ and $Y$ is a translate of…