232 problems

Green–Lazarsfeld secant conjecture. The failure of property $N_p$ is governed by the existence of a $(p+2)$-secant $p$-plane.

Eisenbud–Goto conjecture for selected classes. The conjecture may hold for smooth projective varieties, projectively normal varieties, projective toric varieties, and projective si…

Let $S=k[x_1,[...truncated]$ be a polynomial ring, and let $S^{(c)}$ denote its $c$-th Veronese subring. In zero characteristic, the Green–Lazarsfeld index satisfies … For $n=3$, O…

Let $X$ be a surface and let $A$ be an ample line bundle on $X$. For an integer $p\geq 0$ and an integer $\ell$, set $L=K_X+\ell A$. Mukai's conjecture. The line bundle $L$ satisfi…

Let $[C,\tau]\in\mathcal{R}_{g,\ell}$ be a general level $\ell$ curve of genus $g\geq 5$, where $C$ is a smooth curve and $\tau$ is a non-trivial torsion line bundle. Its paracanon…

Eventual homological linearity conjecture. If $I(G)$ has a linear resolution, then $\operatorname{HS}_i(I(G)^k)$ has a linear resolution for all $i\geq 0$ and all $k\gg 0$.

Let $A$ be an abelian variety defined over an algebraically closed field ${\mathbb K}$. Given an integer $p \geq 0$, an ample line bundle $L$ on $A$, and an integer $m$, the Green–…

Eisenbud–Huneke–Ulrich conjecture. If $I$ is $\mathfrak{m}$-primary and the resolution of $I$ is linear for $p$ steps, where $p\geq 1$, then

Let $S$ be the numerical semigroup under consideration, with multiplicity $m$, generators $d_1,\dots,d_m$, gap power sums $G_r(S)$, and invariants $K_p(S)$. Write…

Let $C$ be a smooth projective curve of genus $g$, let $L$ be a line bundle on $C$, and let $p$ be an integer such that $\deg L\geq 2g+p+1$. Write $\omega_C$ for the canonical line…

Wahl's conjecture. The Gaussian map is surjective when $\mathcal L_1$ and $\mathcal L_2$ are ample.

Let $X$ be an elliptic ruled surface with $e=-1$, and let $L\in \operatorname{Pic}X$ be a line bundle whose numerical class is $aC_0+bf$. Property $N_p$ is the syzygy condition on…

Generic geometric syzygy conjecture. For every $p$, the space of $p$-th linear syzygies of $C$ is spanned by scrollar syzygies.

Let $C \subset \mathbb{P}^{g-1}$ be a general canonical curve of genus $g$. A scrollar syzygy is a $k$th syzygy of rank $k+2$, and the corresponding schemes of such syzygies are ca…

Let $g$ be the genus of a general curve over an algebraically closed field $k$. Eisenbud–Schreyer's conjecture. Generic Green's Conjecture should hold whenever … The source states…

Geometric Syzygy Conjecture. All linear syzygy spaces $\mathrm{K}_{i,1}(C,\omega_C)$ are spanned by syzygies of minimal rank $i+1$.

Let $C$ be a smooth projective curve embedded by a sufficiently positive line bundle $L$, and let $\Sigma_k$ denote its $k^{\text{th}}$ secant variety. Property $N_{k+2,p}$ means t…

Let $X$ be a smooth projective variety, let $B$ be a line bundle on $X$, let $L_d$ be the asymptotic sequence of line bundles considered in the paper, and define the Betti numbers…

Caucci's conjecture. If

Ito–Lozovanu's conjecture. If

Aprodu–Farkas strong maximal rank conjecture. The determinantal variety

Let $R=Q/I$ be a Koszul algebra, and let $t_i(R)$ denote the maximal degree of a minimal generator in homological degree $i$ of the minimal graded free resolution of $R$ over $Q$.…

Let $g\geq 3$, and let $\mathbf{a}$ be the sequence specifying the 2-linear strand of the minimal resolution of a canonical curve. Generic Green's conjecture. There exists a curve…

Let $C$ be a curve of genus $g$ and non-maximal gonality $3\leq k \leq \frac{g+1}{2}$. Assume $W^1_k(C)=\{A\}$ is a reduced single point and $A$ is the unique line bundle of degree…

Let $C$ be a smooth curve of genus $g$, let $p$ be a positive integer, and let $L$ be a non-special line bundle of degree … Assume that $L$ is $(p+1)$-very ample, meaning that it s…