247 problems

Automorphism-group conjecture.

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a smooth projective curve of genus $g$ over $k$, let $B$ be a non-empty set of $r$ points of $X$, and l…

Asymptotic point-count conjecture. For sufficiently large genus $g$, one has

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 $W$ be an irreducible component of the real equiclassical family of curves in $(\mathbb{P}^2)^\vee$ of degree $c$, genus $g$, and class $d$, and let $G$ be a general smooth poi…

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

Let $g$ be a sufficiently large genus, and let a smooth projective curve of genus $g$ have a Jacobian of ST type, as in the source. Coleman's conjecture. Only finitely many smooth…

Let $C$ be a curve over a number field $K$ with an action by a finite group $G$, let $\tau$ be an irreducible representation of $G$, and let $C^\tau$ be the corresponding motivic p…

Oort's conjecture. Every cyclic Galois cover of $k$-curves lifts to characteristic zero.

Let $C$ be a smooth projective curve of genus $g$. For line bundles $A$ and $B$ on $C$ of sufficiently large degree with respect to $g$ and $k$, set $L=A\otimes B$. The $k$-th seca…

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…

The Strong Maximal Rank Conjecture. The locus of line bundles $L\in W^r_dC$ for which $\phi^k_L$ is not of maximal rank has dimension

Let $C$ be a line arrangement that is not a union of concurrent lines, and let ${\alpha}$ be a root of its Alexander polynomial $\Delta^1_C(t)$. Suppose that the order of…

SI-sequence conjecture. The sequence $AV_{X,1}$ is an SI-sequence shifted by $1$. Its last nonzero term is

Set $\rho_{g,k}:=3g-3-\binom{k+1}{2}$. Let $C$ be a general curve of genus $g\geq 2$, and consider the moduli space of stable rank-two vector bundles on $C$ with canonical determin…

Let $k\geq 1$, and let $\overline{\mathcal{C}(1,k)}$ be the projective plane inflectionary curve defined by the homogenized inflection polynomial $P_{1,k}$, with singular points…

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 $\Gamma$ be a set of $L$ irreducible algebraic curves in $\mathbb{C}^n$, each of degree at most $d$, and let $P_2(\Gamma)$ denote the set of points incident to at least two cur…

Strong Oort conjecture. A cyclic-by-$p$ group $G$ is an Oort group for $k$ if and only if $G$ is of the form $C_m$, $D_{2p^n}$, or $A_4$, with the last case occurring only if $p=2$…

Let $C\subset \mathbb{P}^2$ be a plane rational cuspidal curve, meaning a rational plane curve whose singularities are cusps. Orevkov's four-cusp conjecture. The curve $C$ can not…

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 $K$ 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…