122 problems

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…

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

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 $\mu\leq\lfloor n/2\rfloor$,

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 an abelian variety and let $A$ be an ample line bundle on $X$. The M-regularity index $m(A)$ is the largest integer $l$ such that … is $M$-regular for all distinct point…

Let $C$ be a non-hyperelliptic curve with a base-point free tetragonal system $T$, presented as a complete intersection on a three-dimensional rational normal scroll as in the prec…

Let $X$ be a projective toric variety of dimension $n\geq 2$, let $L$ be a line bundle on $X$, and let $p\geq 0$. Property $N_p$ is the syzygy property defined by the minimal grade…

Let $X$ be a projective toric variety of dimension $n$, let $T$ be its algebraic torus, and let $L$ be an ample line bundle on $X$. Let $p$ be a nonnegative integer, and let Proper…

Let $X = \operatorname{\mathbf{Flag}}(p_1|q_1, \dots, p_t|q_t;\;V)$ be a flag supervariety of type $\tau$, and let $Y$ be the variety associated to $X$ by the construction in the p…

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 $\Gamma\subseteq\mathbb{P}^{e}$ be a set of $d$ points in symmetric position. For nonnegative integers $k,m$, say that $\Gamma$ satisfies the property $\propa[k,m]$ if it is no…

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…

Let $X=\mathbb{P}^1\times\mathbb{P}^1$ and let $L=aC_0+bf$. Let $\delta(a,b)$ denote the length of the region of mixed syzygies. Mixed-syzygy length conjecture. The length is … Equ…

Let $X$ be a rational surface and let $L$ be an ample and base point free line bundle on $X$. Suppose that $|L|$ is a linear system containing curves $C$ of genus $g\geq 1$, and wr…

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 $M$ be the coordinate ring of the Veronese, under the hypotheses of Theorem, and let $\beta_{i,i+1}$ denote the corresponding graded Betti numbers. Erman–Martinova conjecture.…