25 problems

Let $X$ be a smooth subvariety of $\mathbb{P}^n$ over $\mathbb{C}$. An Ulrich bundle on $X$ is a nonzero vector bundle satisfying the equivalent linear-resolution, cohomological-va…

Let $Y$ be an Enriques surface and let $H$ be any line bundle on $Y$. Borisov–Nuer's conjecture. There is a line bundle $N\in\operatorname{Pic}(Y)$ such that … This condition is eq…

Let $X$ be a Brauer–Severi variety over a field, with period equal to index. Denote its index by $\operatorname{ind}(X)$ and its categorical representability dimension by…

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 $X\subset\mathcal P^{n+1}$ be a smooth hypersurface of degree $d\geq 2$, and let ${\rm uc}(X)$ denote its Ulrich complexity, namely the minimum rank of an Ulrich bundle on $X$.…

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 $X$ be a projective variety, and let an Ulrich bundle on a projective variety with respect to a very ample line bundle $H$ mean a vector bundle $\mathcal{E}$ satisfying … for e…

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 equip it with the min…

Ulrich rank sufficiency conjecture. The condition $6\mid r(a^2-1)$ should be sufficient for the existence of a rank $r$ Ulrich bundle on…

For integers $d,n\geq 1$, set … and let $X_d^n\subset\mathbb{P}^{N_d^n}$ be the $d$-th Veronese $n$-fold; write $X_d=X_d^3$ when $n=3$. For $d\geq 2$, let $X_d$ be the $d$-th Veron…

Eisenbud–Schreyer's conjecture. The set ${\mathrm{Ur}}(X)$ should always be nonempty. This conjecture concerns the existence of Ulrich sheaves for every polarized projective scheme…

Coskun–Costa–Huizenga–Miró-Roig–Woolf's conjecture. The variety $Fl(k_{1},k_{2};\mathbb{C}^{n})$ does not admit an initialized irreducible homogeneous Ulrich bundle with respect to…

Generic finiteness conjecture. The map $\varphi_{3,d}$ is generically finite onto its image; consequently, its image is a codimension $3$ closed subset of ${\mathcal M}_{2d+4}$.

BGS conjecture. aCM bundles of ranks smaller than the minimal rank of an Ulrich bundle on a smooth hyperquadric do not exist on $X$.

Let $X$ be a smooth Fano threefold of index $2$, and let $r\geq 2$. Denote by $M_X^s(r)$ the moduli space of rank-$r$ stable Ulrich bundles on $X$. Ulrich-bundle moduli conjecture.…

Coskun–Genc conjecture. For every odd integer $d$, there exists a rank-$3$ Ulrich bundle for $(\mathds{P}^2,\mathcal{O}(d))$.

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 $X$ be a variety equipped with a projective embedding, so that Ulrich bundles are defined with respect to that embedding. Ulrich-bundle existence conjecture. Every variety admi…

Ulrich line-bundle conjecture. The surface $S$ admits an Ulrich line bundle.

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 $Y\subset\mathbb{P}^s$ be a smooth hypersurface, and let an aCM or Ulrich vector bundle on $Y$ have rank $r$. The lower-bound conjecture. The rank should satisfy … This conject…

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…

Let $r=2$, and let … be a strictly decreasing sequence of integers divided into three subsequences. For each integer $t$, let $P(t)$ be obtained by subtracting $2t$, $t$, and $0$ f…