247 problems

Let $X$ be a nice curve, meaning a smooth, projective, geometrically integral curve, of genus at least $2$ over $\overline{\mathbb{Q}}$. For a point $x\in X(\overline{\mathbb{Q}})$…

Let $C$ be a nonhyperelliptic curve of genus $g\geq 9$. Let $E\to C$ be a vector bundle of rank $r$ and slope at most $g-2$ such that the tautological line bundle…

Let $(0,0)$ be an isolated singular point of an algebraic curve over a number field $k$, with local ring $\mathcal{O}=k[[x,y]]/(f)$ and multiplicity $m$. Let $C$ be the nonsingular…

Let $C$ be a smooth projective curve of genus $g$. Assume that $C$ is in the generic family. For each $k=1,2,\dots,g$, let $a_k(m;g)$ denote the $m$-th moment of the corresponding…

Let $K$ be an algebraically closed field and let $F$ be an extension field of $K$. Let $S$ be a finite-dimensional $K$-subspace of $F$ such that $K\subset S$. Define the combinator…

Let $X$ be a $Z_n$ curve described by the family in the source. For each positive integer $c$, let $x_c$ be the number of indices $i$ with $c_i=c$, and for each positive integer…

Let $X$ be the $Z_n$ curve associated to the family in the source, with parameters $p$ and $q$, and let $g$ be its genus. A divisor of degree $g$ is non-special when it is not spec…

Let $K$ be an infinite field, let $X$ be a curve over $K$, let $K^{\operatorname{sep}}$ be a separable closure of $K$, and let $G_K$ be the absolute Galois group of $K$. For…

Extension conjecture. Every $\rho\in{\sf Fib}_k(U)$ extends to a functor in ${\sf Fib}_k(V)$.

Strong Main Conjecture. For all $p\notin P_{\mathbf C}$, only finitely many $V\in\mathcal V_p(J)$ give a space $\mathcal H(V\times^s J,\mathbf C)^{\mathrm{in},\mathrm{rd}}$ having…

High-level genus conjecture. For large $k$, all components of $\overline{\mathcal H}_k$ have genus exceeding $1$.

Let $0\leq f<g$, let ${\cal M}_g$ be the moduli space of smooth curves of genus $g$, and let ${\cal H}_g$ be the hyperelliptic locus when $p\geq3$. Write $V_{g,f}$ for the locus of…

Let $V$ be an irreducible variety of dimension $2g+2n-5$, with $4\leq 2n-2<g$, and let $\mathcal{F}\rightarrow V$ be a family of smooth $n$-gonal curves of genus $g$ with maximal v…

Let $X$ and $Y$ be fixed smooth compact complex curves of genus at least $2$, and let the genus of $X$ be $g$. A polynomial bound conjecture. There is a polynomial function $B(g)$…

Simplified presentation conjecture. The presentation has the following properties:

Let $X$ be a projective regular curve over an algebraically closed field $k$, and let $F=k(X)$. Let $\Gamma(F;n)$ and $\Gamma(k;n-1)$ be the weight-$n$ and weight-$(n-1)$ motivic c…

Universality conjecture. Every such curve $C$ is universal.

Let $C'$ be a semistable curve over $F$, and let $C_L$ be the model associated with a finite nonramified extension $L$ of $F(C')$. Geometric Shafarevich conjecture. There is a numb…

Let $K(C)$ be a field as above, and let $\rho$ be an ideal of the field $F^{\mathrm{nr}}$. Strong arithmetic Shafarevich conjecture. There exists a finite extension $F$ of $K$ and…

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 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…

Normalization conjecture for tropical curves. There is an embedding of $C$ with an ideal $I$ having the property that the integral closure of $B$ in its field of fractions is obtai…

Orthogonal splitting type dimension conjecture. Its dimension is

Let $r \geq 1$ and $k \geq 3$. For a general étale double cover $f \colon \widetilde{C} \to C$ in the $k$-gonal locus in $\mathcal{R}_g$, the Prym-Brill-Noether variety $V^r(C,f)$…

Parity conjecture. The unique invariant theta characteristic on $X(p)$ is always even.