32 problems

Let $Y$ be an Enriques surface and let $H$ be any line bundle on $Y$. Borisov–Nuer's conjecture. There is a line bundle $N\in\operatorname{Pic}(Y)$ such that … This condition is eq…

Let $E$ be an Enriques surface, and let $C\subset E$ be an irreducible nonsingular curve of genus $g\geq 2$. Its moduli point in $\overline{\mathcal{M}}_g$ determines a $0$-cycle.…

Let $S$ be an Enriques realizable K3 surface, meaning a K3 surface arising in the Enriques-related moduli setting considered in the source. Bridgeland conjecture. $S$ is Bridgeland…

Let $M=U\oplus U(2)\oplus E_8(-2)$ be the Enriques lattice, let $\mathcal D$ be the relevant orthogonal domain, and let $\mathcal H_{-2}\subset\mathcal D$ be the norm $-2$ Noether–…

The Gromov–Witten potentials involve the generating functions of Gromov–Witten invariants of Enriques and bielliptic surfaces. Quasimodularity conjecture. The Gromov–Witten potenti…

Square-dependence conjecture. If $2\nmid\beta$, then $\mathsf{GV}_{\beta}$ should depend only on the square $\beta^2$.

Enriques perverse Hodge generating-series conjecture. The perverse Hodge numbers satisfy the displayed generating-series identity involving the theta functions, eta functions, and…

Motivic PT generating-series conjecture. The generating series of the fiber-class invariants is given by the displayed infinite-product and plethystic-exponential formula.

The perverse Hodge number generating-series conjecture. The perverse Hodge numbers depend on $\beta$ only through $\beta^2$, and, writing ${^ph}^{i,j}(M_{\beta})={^ph}^{i,j}_d$, th…

Brini–Schüler refined stable-pairs/Gromov–Witten correspondence. One has

Let $Y$ be an Enriques surface and let $v=(r,\beta,n)\in H^{\ast}(Y,\mathbb{Z})$ be primitive, with $r$ even and $2\nmid\beta$. Let $M^Y(v)$ be the automatically smooth moduli spac…

The refined Vafa–Witten multiple-cover conjecture. The refined Vafa–Witten invariant satisfies

Looijenga's conjecture. The smooth Torelli group is trivial:

Let $Y_{158}$ be the realizable $(\tau,\overline{\tau})$-generic Enriques surface associated with the family indexed by $158$, and let $\mathrm{nd}(Y_{158})$ denote its non-degener…

Tanaka–Thomas's conjecture. There exist rational numbers $\mathsf{VW}(v_i)$ such that, for all $n\gg0$,

Klemm–Mariño's formula. For all $g$ and $\beta$,

Let $F$ be a $(k-1)$-very ample bundle of rank $r$ on an Enriques surface. Write … where $v=v(F)=\operatorname{ch}(F)\sqrt{\operatorname{Td}(X)}$ is the Mukai vector. Improved boun…

Let $Y$ be a general Artin–Mumford quartic double solid, let $Y'$ be its blow-up at its $10$ singular points, and let $X$ be the Enriques surface associated to $Y$. Let…

Let $\underline{\Lambda}$ be a Fano $10$-tuple of planes in $\mathbb{P}^5$, and let the objects associated with it be as in Proposition cited in the source, including the EPW sexti…

Let $g=H_Y^2+1\geq 5$ be an odd integer, let $\mathcal F_g$ be the moduli space of polarized $K3$ surfaces of genus $g$, and let $\Sigma_g\subset\mathcal F_g$ be the naturally defi…

Let ${\mathcal E}_{g,\phi}$ be the moduli space of polarized Enriques surfaces $(S,H)$ with $H^2=2g-2$ and … Here $g>1$ and $\phi>0$, and $g$ is the arithmetic genus of curves in…

Let $\overline{M}_{D_{1,6}}^\nu$ be the KSBA compactification of the moduli space of $D_{1,6}$-polarized Enriques surfaces, and let $\overline{\mathcal{D}/\Gamma}^\Sigma$ be the se…

Ulrich line-bundle conjecture. The surface $S$ admits an Ulrich line bundle.

Let $\Lambda=U\oplus \mathbb{E}_8(-1)$ be the even unimodular lattice of signature $(1,9)$, identified with the cohomology lattice of an Enriques surface up to torsion. Difference-…

Unirationality conjecture. The moduli space of polarized Enriques surfaces is always unirational, or at least has negative Kodaira dimension.