18 problems

Derickx–van Hoeij conjecture. There is a modular unit $f$ defined over $\mathbb{Q}$ on $X_1(p)$ such that

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

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 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 $X$ 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 $C$ 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…

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…

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 $C$ 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…

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…

Geometric characterization conjecture. The curve $C$ is $d$-gonерic if and only if

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

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'…