11 problems

Let $C$ be a general curve, and let $(L,V)$ be a general linear system on $C$ with $v\ge2$, where $L$ is a line bundle and $V\subset H^0(L)$ has dimension $v$. Write $D_{L,V}$ for…

Hook-supported extreme coherent-system conjecture. Let $\{M_n\}_n$ be an extreme coherent system on $\mathbb{G}^p$ supported on partitions contained in the hook with $k$ infinite r…

Let $G(\alpha;n,d,k)$ denote the moduli space of $\alpha$-stable coherent systems of type $(n,d,k)$ on the projective line. Suppose that $a\ge2$, $1\le t\le n-1$, and $k\ge n$. Mon…

Non-emptiness conjecture. The space $G(\alpha;n,an-t,k)$ is non-empty for some $\alpha>0$ if and only if one of the following holds:

Let $G(\alpha;r,\mathscr{L}, k)$ be the moduli space of $\alpha$-stable coherent systems of type $(r,d,k)$, where $\mathscr{L}$ is the fixed determinant line bundle and…

Clifford-type bound conjecture. For every $\mu'$-stable coherent system $(\mathcal{O}_S^{\oplus R}\to F)$ with

Let $C$ be a smooth projective curve of genus $g$, let $r,d,k$ be integers, let $\mathcal{L}$ be a line bundle on $C$ of degree $d$, and let $G(\alpha;r,\mathcal{L},k)$ denote the…

Extended Bertram–Feinberg–Mukai conjecture. One has

$n+1$-section nonemptiness conjecture. If

Strong Butler conjecture. The bundle $D_V(E)$ is stable.

A coherent system is a system whose interaction graph has no directed negative loops. In a classical and often-quoted 1981 paper, R. Thomas conjectured that coherent systems do not…