11 problems

Let $\mathbf P^n$ be projective $n$-space, embedded by $\mathcal{O}_{\mathbf P^n}(d)$, and let $\mathrm{K}_{p,1}(\mathbf P^n,\mathcal{O}_{\mathbf P^n}(d))$ denote the Koszul cohomo…

Let $\mathbf P^n$ be projective $n$-space, embedded by $\mathcal{O}_{\mathbf P^n}(d)$, and let $\mathrm{K}_{p,q}(\mathbf P^n,\mathcal{O}_{\mathbf P^n}(d))$ denote the corresponding…

The Geometric Syzygy Conjecture in Even Genus. The last linear syzygy space

Precise nonvanishing range conjecture. In the situation of that theorem, one has

Asymptotic vanishing conjecture. Fix $q\in[2,n]$. In the situation of the nonvanishing theorem referenced in the source, there is a constant $C_3$, depending on $X$, $A$, $B$ and…

Let $X$ be a smooth projective variety of dimension $n$, let $L_d=dA+E$ with $A$ ample and $E$ an arbitrary divisor on $X$, and let $B$ be a line bundle on $X$. For the Koszul coho…

Ottaviani–Paoletti conjecture.

Farkas's conjecture. The curve $C$ has a pure resolution, equivalently it satisfies property $N_p$.

Minimal Resolution Conjecture. Fix integers $g,r,d,p\geq1$ such that $g-d+r\geq0$, and assume

Minimal Syzygy Conjecture. One has

Let $C$ be a general curve of genus $g=2d-2\geq6$, let $\eta\in\operatorname{Pic}^0(C)$ be a general line bundle, and let $K_{p,q}(C,L)$ denote Koszul cohomology. Green-Lazarsfeld'…