34 problems

Let $C$ be a smooth curve, let ${\cal L}$ be a line bundle on $C$, and let $V\subseteq H^0(C,{\cal L})$ define a linear series. Write $M_{V,{\cal L}}$ for the dual span bundle asso…

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…

Brill–Noether conjecture. If $\rho<0$, then $C$ carries no line bundle $A$ of degree $d$ with $h^0(A)\geq r+1$.

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 $f:C\to\mathbb{P}^1$ be a general degree-$k$ cover of a genus-$g$ curve, and let $W^{\vec e}C$ be the splitting-type Brill–Noether locus with splitting type $\vec e$. Let $r$ d…

Let $g\geq 3$. For integers $r,d$, let $\mathcal{M}^{r}_{g,d}\subseteq\mathcal{M}_g$ denote the locus of smooth curves of genus $g$ admitting a $g^r_d$, and consider the expected m…

Asher–Haburcak conjecture. When $g\ne 7,8,9$, the locus ${\mathcal M}^r_{g,d}$ is maximal: it is not contained in any other locus ${\mathcal M}^{r'}_{g,d'}$. These loci are the max…

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 $C$ be a smooth projective curve, let $L$ be a line bundle on $C$, and let $V\subseteq H^0(C,L)$ be a subspace such that $(L,V)$ is a generating pair, meaning that $V$ generate…

Let $C$ be a smooth projective curve over $\mathbb{C}$, let $L$ be a globally generated line bundle on $C$, and let $V=H^0(C,L)$, so that $|L|$ is the complete linear system and…

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 $C$ be an algebraic curve degenerating to a metric graph $\Gamma$, and let a rank-$r$ map have image on $\Gamma$. A degeneration conjecture is that, in general, this image is t…

Classical maximal rank conjecture. The map should always be injective or surjective, and the analogous assertion should hold for the higher-order multiplication maps.

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 general curve of genus $g$, and let $W^r_d(C)$ be the variety of linear systems of degree $d$ and dimension at least $r$. For positive integers $r,d$ satisfying … wher…

Eisenbud–Harris maximal rank conjecture. The map $\phi_L$ has maximal rank.

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

Quadratic maximal rank prediction. The maximal rank conjecture predicts that $\mu_2$ is injective for a general linear series on a general curve exactly when $r\leq 2s$.

Tropical maximal rank conjecture. Suppose $r\geq 3$, $\rho(g,r,d)\geq 0$, and $d<g+r$. Then there is a divisor $D$ of rank $r$ and degree $d$ whose class is vertex avoiding on a ch…

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

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 of genus $g$, let $W^1_d(C)$ denote the variety parametrizing complete linear series of degree $d$ and dimension one on $C$, and let…

Nakamaye–Lang conjecture. The subgroups $H_i$ should appear as the base locus of a linear series as in Theorem 1.1, in the sense that the base locus is described by unions of trans…