123 problems

For a general smooth curve $C$ of genus $g\ge 2$ and a nontrivial $2$-torsion line bundle $\eta\in \operatorname{Pic}^0(C)[2]$, the Prym-canonical line bundle $K_C\otimes\eta$ sati…

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 $X$ be a curve of genus $g$. For a divisor $D$ on $X$ of degree $d$, let $g_d^r$ denote a subspace of the Riemann–Roch space $L(D)$ of dimension $r+1$, and let $\beta_{p,2}$ be…

Prym-Green conjecture. The resolution of a general level $\ell$ paracanonical curve of genus $g$ is natural.

Let $C$ be a curve of gonality $d$, and let $L$ be a nonspecial very ample line bundle on $C$. Writing $h^0(L)$ for the dimension of its space of global sections, Green–Lazarsfeld'…

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

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 $X$ 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…

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

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 $C$ 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…

Ottaviani–Paoletti conjecture.

Let $v_k(\mathbb{P}^d)$ be the $k$-uple Veronese embedding of $\mathbb{P}^d$, with $d\geq2$ and $k\geq3$, and let $\Sigma$ be its secant variety. Write $N_{p,q}$ for the indicated…

Let $X$ be a smooth curve embedded by a line bundle $L$, and let $Sec^{k-1}X$ denote its $(k-1)$-st secant variety. Secant-variety degree-bound conjecture. If $\deg(L)\geq 2g+2k$,…

Let $X$ be a smooth variety, let $L$ be an ample line bundle on $X$, and fix an integer $k\geq 1$. For sufficiently large $n$, the line bundle $L^n$ defines an embedding of $X$; wr…

Let $X$ be a smooth curve and let $L$ be a very ample line bundle defining an embedding of $X$. For each integer $k$, let $Sec^kX$ denote the $k$-th secant variety. Eisenbud–Koh–St…

Generic Green's conjecture. The generic curve $C$ of genus $g$ satisfies

Let $X$ be an abelian variety, let $A$ be an ample line bundle on $X$, and let $m(A)$ denote its index of $M$-regularity. For non-negative integers $p\ge m$, Pareschi–Popa's syzygy…

Let $k[\mathbf{x}]=k[x_1,\ldots,x_n]$, and let $F$ be a finite set of $q\geq n$ monomials of the same degree $d\geq 2$. Suppose that the ideal $(F)\subset k[\mathbf{x}]$ is of resi…

Let $\mathcal{M}_g^0$ be the open moduli space under consideration, and let $\mathcal{Z}_{g,i}$ be the locus of curves $C$ for which there exists a pencil $A\in W^1_k(C)$ such that…

Let ${\mathcal A}$ be a complex hyperplane arrangement, let $A=H^*(X({\mathcal A}),\Bbbk)$ be its Orlik-Solomon algebra over a field of characteristic $0$, and let $E$ be the exter…

Let $G$ be a finite group in the nonmodular case acting on a finite-dimensional representation $V$. Let $\beta_G^i(V)$ denote the degree bound for the $i$-th syzygies of the invari…

Mu-class closure conjecture. For every $\mu\leq\lfloor n/2\rfloor$,