21 problems

Bondal's conjecture. The locus $\mathsf{D}_{2k}(\sigma)$ has an irreducible component of dimension at least $2k+1$ for every such integer $k$. The conjecture predicts that degenera…

Tropical Brill–Noether cycle conjecture. Assume $\rho\geq 0$. Then there exists a canonical tropical subvariety $Z^r_d\subseteq W^r_d$ of pure dimension $\rho$ such that

Let $g\geq 1$ be an integer, let $\pi\colon\mathfrak{A}_g\to\mathbb{A}_g$ be the universal family of principally polarized abelian varieties with suitable level structure, and let…

The tableau interpretation conjecture. The coefficient $c_\mu(r)$ equals the number of sequences of tableaux satisfying the conditions specified by the quiver-coefficient construct…

Let $\mathcal{M}_g^0$ be the open moduli space under consideration, and let $\mathcal{Z}_{g,i}$ be the locus of curves $C$ for which there exists a pencil $A\in W^1_k(C)$ such that…

Fulton–Lazarsfeld's connectivity conjecture. If

Göpel's orbital degeneracy-locus conjecture. Let $v$ be a general element of $\Delta$. Then:

Let $g\in\mathbb{N}$, and let $X$ be an irreducible closed subvariety of the universal family $\mathfrak{A}_g$ defined over $\overline{\mathbb{Q}}$. Write…

Let $m\geq1$, let $s\in\{m+3,\ldots,2m+3\}$, and set $n=2s-2m-3$. Let $\mathsf{M}_{m,s}$ be the degeneracy locus considered in the paper and let $Z_{n,s,m}$ be the associated varie…

Hilbert-scheme realization conjecture. The variety $Z$ can be realised as a Hilbert scheme on $Y$. This is proposed as a further relation between $Z$ and the Hilbert scheme, and th…

Let $m\geq 0$, $n\geq 3$, and $s\in\{m+2,\ldots,2m+3\}$. Let $S$ be the first degeneracy locus of a general morphism … and let $Z$ be the associated zero locus in…

Let $A_1,\dots,A_{m+1}$ be linear operators on ${\mathbb R}^m$, and let ${\mathcal X}$ be their degeneracy locus in ${\mathbb P}^{m-1}({\mathbb R})$. The locus has expected dimensi…

Let $A_1,\dots,A_5$ be real matrices whose degeneracy locus ${\mathcal X}:={\mathcal X}(A_1,\dots,A_5)$ is formed from the associated ten lines in ${\mathbb P}^3$. Desargues config…

Let $C$ be a smooth curve, let $p,q$ be marked points on $C$, and consider the flags of subspaces obtained by recording vanishing orders at $p$ and $q$ as the line bundle varies in…

Let $\Sigma^\wedge_{\infty,i}$ and $\Sigma^S_{\infty,i}$ denote the stable skew-symmetric and symmetric degeneracy loci, and write their Segre–Schwartz–MacPherson classes as formal…

Let $E$ be a vector bundle in the setting above, let $\overline{\Sigma}_{\vec{e}}(E)$ denote the splitting degeneracy scheme associated to the splitting type $\vec{e}$, and suppose…

Let $X$ be a smooth complex compact variety of dimension $n$ and let $\pi$ be a Poisson structure on it. For an integer $k\geq 0$ satisfying $2k<n$, define the degeneracy locus…

Peskine's conjecture. The Palatini scroll is the only smooth and non-quadratically normal threefold in $\mathbb{P}^5$.

Let $X$ be the threefold in $\mathbb{P}^{5}$ obtained as the degeneracy locus of a general morphism $\mathcal{O}_{\mathbb{P}^{5}}^{4}\to\Omega_{\mathbb{P}^{5}}(2)$. It is a scroll…

Optimistic Poisson degeneracy conjecture. If $X_s$ is nonempty, then it contains a component of dimension greater than $s$.

Let $C$ be a general curve, let $L$ be a line bundle on $C$, and let $n$ be a positive integer. Consider the natural multiplication map … Its degeneracy loci are the loci where thi…