464 problems

Green–Lazarsfeld secant conjecture. The failure of property $N_p$ is governed by the existence of a $(p+2)$-secant $p$-plane.

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

Let $X/k$ be a curve with Jacobian $J$, and write $g(X)=\dim J$. A CM Jacobian is a Jacobian with complex multiplication. Degeneracy can only occur when … Coleman's conjecture. Up…

Let $C$ be an integral curve of genus $g$, and for each positive integer $n$ define … Here ${\rm Cliff}_1(C)$ is the rank-one Clifford index. Mercat's conjecture. … The conjecture…

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…

Expected-dimension conjecture. If $\operatorname{codim}_{\mathcal{M}_g}\mu(\mathcal{H})\le g-5$ and $\mathcal{X}(d,g,r)\ge 0$, then $\mathcal{H}$ has the minimal possible dimension…

Abhyankar's conjecture. The finite quotient groups of the étale fundamental group of $\mathbb{G}_m$ are precisely the finite groups $G$ such that

Let $k$ be a field of characteristic $p>0$, and let a cyclic extension mean a finite Galois extension with cyclic Galois group. Oort's conjecture. Every cyclic extension lifts to c…

Grothendieck's section conjecture in arbitrary characteristic. The natural morphism

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…

Strong Maximal Rank Conjecture. The locus $Q^r_d(C)\subset G^r_d(C)$ consisting of linear series for which $\nu(L,V)$ fails to have maximal rank is a determinantal locus of the exp…

An optimal recursive tower is a recursive tower of algebraic function fields attaining the Drinfeld–Vlăduț bound. Elkies' conjecture. Every optimal recursive tower is modular in an…

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 $d$ be an integer, and let $C$ be an algebraic curve of degree $d$ in $\mathbb{C}P^2$. Its genus is $\frac{(d-1)(d-2)}{2}$. Thom's conjecture. The genus of $C$ is minimal among…

Let $(S,H)$ be a polarized K3 surface, and suppose that the complete linear system $|H|$ is base point free. For smooth curves $C$ in $|H|$, the gonality is the least integer $k$ f…

Brill–Noether conjecture. For general $C$, the locus $W^r_dC$ is nonempty if and only if

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…

Let $\mathscr{H}_{k,g}$ be the Hurwitz stack parametrizing degree $k$ covers by genus $g$ curves of $\mathbb{P}^1$, up to automorphisms of the target $\mathbb{P}^1$, and let…

Let $C$ be a smooth curve, and write $g$ for its genus, $\operatorname{gon}(C)$ for its gonality, and $\operatorname{Cliff}(C)$ for its Clifford index. Curves satisfying…

Let $g$ be the genus of a general curve over an algebraically closed field $k$. Eisenbud–Schreyer's conjecture. Generic Green's Conjecture should hold whenever … The source states…

Let $k\geq 3$, let $g\geq 0$, and consider the polytope … A scrollar-invariant tuple is a tuple of positive integers $\mathbf e=(e_1,\dots,e_{k-1})$ with $e_1+\dots+e_{k-1}=g+k-1$.…

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

Let $X$ be an integral regular projective curve over $\Bbb F_q$ of genus $g$, and let $n$ be the rank parameter. The rank-$n$ zeta function is denoted by…

Let $E$ and $E'$ be $p$-adic fields, and let $Y/E'$ and $X/E$ be anabelomorphic, geometrically connected, smooth hyperbolic curves. Here, anabelomorphic means that their associated…