8 problems

Asymptotic semistability conjecture. There exists an integer $m_0\gg0$ such that, for every $m\ge m_0$, the syzygy bundle $S_{\mathcal{E}(m)}$ is semistable. This is proposed as a…

Let $X\subset \mathbb{P}^{N}$ be a smooth projective variety, and call a vector bundle on $X$ Ulrich when it satisfies the Ulrich conditions for the given embedding. Ulrich's conje…

Let $(X,L)$ be a polarized pair, meaning a projective variety $X$ equipped with a polarization $L$, and let an Ulrich bundle mean a vector bundle satisfying the Ulrich conditions w…

Let $(\mathbb{P}^2,dH)$ denote the projective plane polarized by $dH$, where $d$ is an odd integer and $d\geq 3$. Rank 3 Ulrich-bundle conjecture. There exists a rank $3$ Ulrich bu…

Let $\Lambda=U\oplus \mathbb{E}_8(-1)$ be the even unimodular lattice of signature $(1,9)$, identified with the cohomology lattice of an Enriques surface up to torsion. Difference-…

Let $(\alpha,\beta,\gamma)$ be the type of an Ulrich partition. Nonexistence conjecture. There are no Ulrich partitions of type $(\alpha,\beta,\gamma)$ if $\beta\geq3$ and…

Let $(\alpha,\beta,\gamma)$ be the type of an Ulrich partition, and call two types symmetric when they are identified by the symmetry used in the paper. Here $n$ is a positive inte…

Let $F(k_1,k_2;n)$ be the two-step flag variety parametrizing flags of subspaces of dimensions $k_1$ and $k_2$ in an $n$-dimensional vector space, and let $\mathcal{O}(1)$ denote t…