27 problems

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…

Dimension-equality conjecture. For every $g\geq 0$ and $d\geq 1$,

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…

Clifford-dimension conjecture. One has

Let $X$ be a smooth projective curve of genus $g$, let $k$ be a positive integer, and let ${\cal M}_{g,k}$ denote the moduli space of $k$-gonal curves of genus $g$. Assume…

Let $C$ be a smooth irreducible projective curve, with gonality $\operatorname{gon} C$ and Clifford index $\operatorname{Cliff} C$. Curves satisfying … are called exceptional curve…

Let $f$ be a non-isotrivial one-parameter algebraic family of endomorphisms of ${\mathbb P}^1$ over ${\mathbb C}$, and let $(\Gamma_n)$ be a sequence of distinct dynatomic curves.…

Let $P$ be a lattice polygon and let $\Gamma$ be a smooth tropical plane curve with Newton polygon $P$. The expected-gonality conjecture asserts that … Here…

Gonality-tight graph conjecture. The following hold:

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 $X\subseteq\mathbb{P}^{n+r}$ be a very general smooth complete intersection of dimension $n>0$, cut out by polynomials of degrees $d_{1},\ldots,d_{r}$, and let…

Gonality conjecture. The gonality of $C$ can be detected from the syzygies of $C$ in the embedding defined by $L$.

Choi–Kwak gonality-sequence conjecture. One has

Let $C$ be a Gorenstein unicuspidal rational curve of genus $g$, and let $\gon_F(C)$ denote its covering gonality, namely the smallest degree of a map from $C$ to $\mathbb{P}^1$. C…

Green–Lazarsfeld Gonality Conjecture. The moment when the first zero entry appears in row $1$, namely column $b+1$, is determined by the gonality $\operatorname{gon}(C)$ of the cur…

Let $q$ be a prime power, let $\gamma$ be a positive integer, and let $g$ be a positive integer. Let $N_q(g,\gamma)$ denote the maximal number of $\mathbb F_q$-rational points on a…

Optimal Gonality Conjecture. Every optimal curve $C$ over $\mathbb{F}_q$ of genus $g$ has gonality at most $\operatorname{geomgon}$.

Gonality-point equality conjecture. Fix $\gamma\geq 2$. For $g$ sufficiently large,

The splitting type stratification conjecture.

Let $k\in\mathbb{Z}_{>0}$, and let $A$ be a very general abelian variety. Its gonality is the least degree of a nonconstant map from a smooth projective curve to $A$. Voisin's gona…

Let $C\subset \mathbb{P}^{g-1}$ be a smooth canonically embedded curve of genus $g\neq 6$, and let $k<\lceil g/2\rceil$. Denote by $W^1_k(C)$ the scheme of degree-$k$ pencils on…

Let $C$ be a curve of gonality $3\leq d\leq\frac{g+1}{2}$. Assume that $C$ has a unique minimal pencil $f:C\to\mathbb{P}^1$, that the Brill--Noether locus … is reduced, and that…

Let $X\subset \mathbb{P}^{n+1}$ be a very general hypersurface of degree $d\geqslant 2n$. Here $\operatorname{cov.gon}(X)$ denotes the covering gonality of $X$, namely the least go…

Let $C$ be a smooth complex algebraic curve of genus $g$ and gonality $k$, and let $L$ be a very ample line bundle on $C$ with … Write $h^0(L)=\dim H^0(C,L)$, and let…

Gonericity conjecture. In characteristic zero, $C$ is $\ell+2$-goneric if and only if