This is a brief introduction to the theory of Enriques surfaces over arbitrary algebraically closed fields.

This survey paper is about moduli spaces in algebraic geometry for which a period map gives that space the structure of a (possibly incomplete) locally symmetric variety and about their natural compactifications. We outline the Baily-Borel compactification for such varieties, and show that it usually differs from the compactifications furnished by the standard techniques in algebraic geometry. It turns out however, that a reconciliation is possible by means of a generalization of the Baily-Borel construction for the class of incomplete locally symmetric varieties that occur here.

The emphasis is here on moduli spaces of varieties other than that of polarized abelian varieties.

We survey classical and recent developments in the theory of moduli spaces of sheaves on projective varieties.

The moduli space $\mathcal{M}_g$ of nonsingular projective curves of genus $g$ is compactified into the moduli $\overline{\mathcal{M}}_g$ of Deligne-Mumford stable curves of genus $g$. We compactify in a similar way the moduli space of abelian varieties by adding some mildly degenerating limits of abelian varieties.

A typical case is the moduli space of Hesse cubics. Any Hesse cubic is GIT-stable in the sense that its $\mathrm{SL}(3)$-orbit is closed in the semistable locus, and conversely any GIT-stable planar cubic is one of Hesse cubics. Similarly in arbitrary dimension, the moduli space of abelian varieties is compactified by adding only GIT-stable limits of abelian varieties (§ 14).

Our moduli space is a projective “fine” moduli space of possibly degenerate abelian schemes with non-classical non-commutative level structure over $\mathbf{Z}[\zeta_{N},1/N]$ for some $N\geq 3$. The objects at the boundary are singular schemes, called PSQASes, projectively stable quasi-abelian schemes.

In this short note, we extend the results of [Alexeev-Orlov, 2012] about Picard groups of Burniat surfaces with $K^2=6$ to the cases of $2\le K^2\le 5$. We also compute the semigroup of effective divisors on Burniat surfaces with $K^2=6$. Finally, we construct an exceptional collection on a nonnormal semistable degeneration of a 1-parameter family of Burniat surfaces with $K^2=6$.

We introduce a new big $I$-function for certain GIT quotients $W/\!\!/\mathbf{G}$ using the quasimap graph space from infinitesimally pointed $\mathbb{P}^1$ to the stack quotient $[W/\mathbf{G}]$. This big $I$-function is expressible by the small $I$-function introduced in [6, 10]. The $I$-function conjecturally generates the Lagrangian cone of Gromov-Witten theory for $W/\!\!/\mathbf{G}$ defined by Givental. We prove the conjecture when $W/\!\!/\mathbf{G}$ has a torus action with good properties.

We study irreducibility of families of degree 4 Del Pezzo surface fibrations over curves.

In this note we interpret a recent result of Gaberdiel et al [7] in terms of derived equivalences of K3 surfaces. We prove that there is a natural bijection between subgroups of the Conway group $C\!o_0$ with invariant lattice of rank at least four and groups of symplectic derived equivalences of $\mathrm{D}^{\mathrm{b}}(X)$ of projective K3 surfaces fixing a stability condition.

As an application we prove that every such subgroup $G\subset C\!o_0$ satisfying an additional condition can be realized as a group of symplectic automorphisms of an irreducible symplectic variety deformation equivalent to $\mathrm{Hilb}^n(X)$ of some K3 surface.

In each characteristic $p\neq 2, 5$, it is shown that order 40 automorphisms of K3 surfaces are purely non-symplectic. Moreover, a K3 surface in characteristic $p\neq 2, 5,$ with a cyclic action of order 40 is isomorphic to the Kondō's example.

This is a continuation of our paper [13] where we considered markings of Kählerian K3 surfaces by Niemeier lattices and studied in details these markings and their applications to finite symplectic automorphism groups and non-singular rational curves of K3 surfaces for the first Niemeier lattices $N_k$, $k=1,2,\dots,21$.

Here we do the same for the remaining Niemeier lattices $N_{22}$ and $N_{23}$ with root systems $12A_2$ and $24A_1$ respectively.

We shall study the chamber structure of positive cone of the albanese fiber of the moduli spaces of stable objects on an abelian surface via the chamber structure of stability conditions.