241 problems

Gopakumar–Vafa integrality conjecture. The BPS invariants $n_{h,\beta}(X)$ defined by this equation should be integers for all $h\geq 0$ and $\beta\neq 0$.

Stable-pair rationality and symmetry conjecture. The series $Z_\beta^\mathsf{PT}(q\mid D)$ is the Laurent expansion of a rational function, more precisely of $q^{d_\beta/2}f(q)$ fo…

Let $X$ be a surface, let $\mathcal{L}$ be a line bundle on $X$, and let $\delta$ be a non-negative integer. Let $N_\delta^X(\mathcal{L})$ be the number of curves in…

Gromov–Witten/Donaldson–Thomas correspondence. The GW theory and the DT theory of $X$ should be equivalent after a change of variables.

Let $X\subset \mathbb{P}^N$ be a smooth Fano hypersurface of degree $d$, and let $F(X)$ denote its Fano scheme of lines. The expected dimension is $2N-d-3$. Debarre–de Jong conject…

Pandharipande's Gopakumar--Vafa integrality conjecture. The primary Gromov--Witten theory of threefolds is governed by certain integers, called Gopakumar--Vafa invariants or BPS in…

Let $f:{\mathbb{C}}^n\to {\mathbb{C}}^m$ be a map, and let its Thom polynomials be expressed in the Chern classes of $f$. Rimányi's Chern positivity conjecture. These Thom polynomi…

Mori cone generation conjecture. The Mori cone $M$ of $X_\psi$ is generated by the classes of the curves $\ell_{\mathbf{t}}$, $\sigma_{\mathbf{s},\mathbf{t}}$, and…

Let $X$ be the CHL geometry in the source, let $C'(\lambda)\subset X$ be the subscheme associated with a partition $\lambda$, and let $\operatorname{Hilb}^m_{C(\lambda)}$ be the co…

Let $X$ be a nonsingular projective $3$-fold. The Gromov–Witten theory of $X$ consists of its Gromov–Witten invariants, and the Donaldson–Thomas theory consists of its Donaldson–Th…

Lehn's conjecture. The generating series satisfies

Degree-zero DT conjecture. The degree-zero Donaldson–Thomas partition function is

Let $g\geq 2$ be the genus, let ${\mathsf{r}}=2$ be the rank, and let ${\mathsf{d}}$ be the degree. Define the generating series $\check{\mathsf{QM}}^i_{\mathsf{d}}(q)$ and…

Yau–Zaslow conjecture. The generating function for rational curves with $n$ double points on a K3 surface is

Let $X$ be a nonsingular projective 3-fold with only $(p,p)$-cohomology, let $\beta\in H_2(X,\mathbb{Z})$, and let $\mathbb{D}^X_{\mathrm{PT}}$ be the stable pairs descendent algeb…

Assume that $X$ is rational, $D$ is a smooth anticanonical curve on $X$, and $P\in D(\beta)$, where $D(\beta)$ is the finite set of points associated with the class $\beta$. Let…

LMOV conjecture. For every $\mu\in\mathcal{P}_+$, the function $F_{\mu}^{(X,\mathcal{D})}$ has the integral expansion given by the right-hand side of the open multiple-covering for…

For a smooth projective plane, let $N^{d,\delta}$ denote the Severi degree, namely the number of possibly reduced $\delta$-nodal degree-$d$ curves passing through general…

Let $(X,L)$ be a very general polarized $K3$ surface. A rational curve in $|L|$ is a curve of geometric genus $0$. Nodal rational curves conjecture. All rational curves in $|L|$ ar…

Di Francesco–Itzykson conjecture. For fixed $delta$, the Severi degree $N^{d,delta}$ is equal to the node polynomial $N_delta(d)$ for all sufficiently large $d$.

Let $X$ be a smooth projective three-fold. Write $Z_{DT}(X,q)_0$ for the degree-zero Donaldson–Thomas generating series, let $T_X$ be the tangent bundle, let $K_X$ be the canonical…

The paper considers a suitably general algebraic system of curves on an algebraic family of surfaces. In the enumerative setting, the relevant enumerative numbers are encoded by a…

Fix a degree $d$ and geometric genus $g$, and let the normalization of the Severi variety parametrize plane curves with these invariants. Let $A$, $B$, and $C$ be the divisor class…

Let $A$, $B$, and $C$ be the standard divisor classes on the Severi variety, and let the boundary divisors denote the divisors parametrizing reducible or otherwise degenerate curve…

Götsche's conjecture. Certain quasi-modular forms are the generating functions for the number of curves in the fixed linear system $|C|$.