20 problems

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 $X\to B$ be a general family of genus $g$ stable curves whose general member has Clifford index $\mathfrak{c}$, and let $B$ be a general curve in $\overline{\mathcal D}^r_d$. C…

Let $C$ be a non-hyperelliptic curve of genus $g$ with a base-point free tetragonal system $T$, and suppose $C$ has the complete-intersection presentation on a three-dimensional ra…

Let $C$ be an irreducible projective smooth complex curve, let $L$ be a line bundle on $C$, and let $V\subseteq H^0(C,L)$ define a base point free non-complete linear series. Let…

Mistretta–Stoppino's conjecture. If

Ito–Lozovanu conjecture. Under these assumptions, $L$ satisfies property $(N_p)$, meaning that the first $p$ steps of the minimal graded free resolution of its section algebra are…

Let $C$ be a smooth projective curve and let $L$ be a line bundle such that $V=H^0(C,L)$ gives a complete base point free linear series. Write $|L|$ for this complete linear series…

Let $X$ be a smooth curve of genus $g$, and let an injective linear series $g^2_d$ on $X$ mean a base-point-free, not necessarily complete, two-dimensional linear series of degree…

Let $g$, $r$, and $d$ be positive integers such that $g-d+r\geq 0$ and $0\leq \rho(g,r,d)\leq r-2$. For a general curve $X$ of genus $g$, let $W^r_d(X)$ denote the variety of divis…

Let $C$ be a curve, let $L$ be a globally generated line bundle on $C$, and let $M_L$ be the kernel bundle in the evaluation sequence … The pair $(H^0(C,L),L)$ is linearly (semi)st…

Maximal Rank Conjecture. The rank of this restriction map is

Maximal rank conjecture. The multiplication maps $\mu_m$ have maximal rank for all $m$.

Noether's maximal rank conjecture. The maps $\mu_m$ have maximal rank for all $m$; that is, each $\mu_m$ is either injective or surjective. The conjecture is a central open problem…

Let $S$ be a $K3$ surface, let $C$ be a smooth irreducible curve of genus $g\geq 2$ on $S$, and let $A$ be a complete, base point free linear series $g^r_d$ on $C$, with…

Let $C$ be a smooth curve and let ${\cal L}$ be a line bundle on $C$. Let $M_{\cal L}$ denote the kernel bundle of the evaluation map…

Let $X$ be the blow-up of $\mathbb P^2$ at $r$ general points, let $D$ be an effective divisor on $X$, and let $E$ be a $(-1)$-curve. Consider the restriction map … The conjecture…

Let $X$ be a smooth projective surface that is either rational or defined in characteristic zero. The conjecture asserts that there exists a constant $c_X$ such that every prime di…

Let $S=\mathbb P^2$, let $I=(y,x^r)^m$ with $r$ and $m$ natural numbers and $r>9$, and let $Y$ be a general member of ${\rm Hilb}_I S$. For a divisor $D=dH$ on $\mathbb P^2$, the m…

Let $S=\{p_1,\dots,p_r\}$ be distinct points in $\mathbb P^N$ over an algebraically closed field, and let $a(S,m)$ and $e(S)$ be defined by … Chudnovsky's conjecture, proved for…

Minimal Syzygy Conjecture. One has