Divisors on Burniat surfaces

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 $K^2=6$.Comment: v.2: small correction for the $K^2=2$ cas

Log canonical singularities and complete moduli of stable pairs

1) Assuming log Minimal Model Conjecture, we give a construction of a complete moduli space of stable log pairs of arbitrary dimension generalizing directly the space M_{g,n} of pointed stable curves. Each stable pair has semi log canonical singularities. 2) We prove that a stable quasiabelian pair, defined by author and I.Nakamura as the limit of abelian varieties with theta divisors, has semi log canonical singularities.Comment: AMSLaTeX 1.2/LaTeX2e, Postscript file is also available at http://domovoy.math.uga.edu/preprint

Complete moduli in the presence of semiabelian group action

I prove the existence, and describe the structure, of moduli space of pairs $(p,\Theta)$ consisting of a projective variety $P$ with semiabelian group action and an ample Cartier divisor on it satisfying a few simple conditions. Every connected component of this moduli space is proper. A component containing a projective toric variety is described by a configuration of several polytopes, the main one of which is the secondary polytope. On the other hand, the component containing a principally polarized abelian variety provides a moduli compactification of $A_g$. The main irreducible component of this compactification is described by an "infinite periodic" analog of the secondary polytope and coincides with the toroidal compactification of $A_g$ for the second Voronoi decomposition.Comment: 98 pages, published versio

Higher-dimensional analogues of stable curves

The Minimal Model Program offers natural higher-dimensional analogues of stable $n$-pointed curves and maps: stable pairs consisting of a projective variety $X$ of dimension $\ge2$ and a divisor $B$, that should satisfy a few simple conditions, and stable maps $f:(X,B)\to Y$. Although MMP remains conjectural in higher dimensions, in several important situations the moduli spaces of stable pairs, generalizing those of Deligne-Mumford, Knudsen and Kontsevich, can be constructed more directly, and in considerable generality. We review these constructions, with particular attention paid to varieties with group action, and list some open problems

Limits of stable pairs

Let (X_0,B_0) be the canonical limit of a one-parameter family of stable pairs, provided by the log Minimal Model Program. We prove that X_0 is S2 and that [B_0] is S_1, as an application of a general local statement: if (X,B+\epsilon D) is log canonical and D is Q-Cartier then D is S2 and [B] \cap D is S1, i.e. has no embedded components. When B has coefficients smaller than 1, examples due to Hacking and Hassett show that B_0 may indeed have embedded primes. We resolve this problem by introducing a category of stable branchpairs. We prove that the corresponding moduli functor is proper for families with normal generic fiber.Comment: 13 page

Moduli spaces Mg,n(W)M_{g,n}(W) for surfaces

We construct and prove the projectiveness of the moduli spaces which are natural generalizations to the case of surfaces of the following: 1) $M_{g,n}$, the moduli space of $n$-marked stable curves, 2) $M_{g,n}(W)$, the moduli space of $n$-marked stable maps to a variety $W$.Comment: 21 pages, written in AMS-LaTe

Boundedness and K2K^2 for log surfaces

Let $\epsilon, C$ be two positive real numbers, and $\mathcal C \subset \mathbb R$ be a DCC (descending chain condition) set. Let $(X, B = \sum b_j B_j)$ denote a projective surface with an $\mathbb R$-divisor. Then (1) The class $\{X\}$ of surfaces for which there exists a divisor $B$ such that $(X,B)$ is $\epsilon$-log terminal and $-(K_X + B)$ is nef (excluding only those for which at the same time $K_X\equiv 0$, $B=0$, and $X$ has at worst Du Val singularities), is bounded. (2) The set $\{(K_X + B)^2\}$ of squares for the semi log canonical pairs $(X, B)$ with ample $K_X + B$ and $b_j \in \mathcal C$, is a DCC set. (3) The class $\{(X,B)\}$ of pairs such that $(X, B)$ is semi log canonical, $K_X + B$ is ample, $(K_X + B)^2 = C$ and $b_j \in \mathcal C$, is bounded.Comment: This version: a TeX fix only. The old TeX version did not work with pdflatex, producing strange character

Weighted grassmannians and stable hyperplane arrangements

We give a common generalization of (1) Hassett's weighted stable curves, and (2) Hacking-Keel-Tevelev's stable hyperplane arrangements.Comment: 19 pages. v2: misc minor improvement

Two Two-dimensional Terminations

Varieties with log terminal and log canonical singularities are considered in the Minimal Model Program, see \cite{...} for introduction. In \cite{shokurov:hyp} it was conjectured that many of the interesting sets, associated with these varieties have something in common: they satisfy the ascending chain condition, which means that every increasing chain of elements terminates. Philosophically, this is the reason why two main hypotheses in the Minimal Model Program: existence and termination of flips should be true and are possible to prove. In this paper we prove that the following two sets satisfy the ascending chain condition: 1. The set of minimal log discrepancies for $K_X+B$ where $X$ is a surface with log canonical singularities. 2. The set of groups $(b_1,...b_s)$ such that there is a surface $X$ with log canonical and numerically trivial $K_X+\sum b_jB_j$. The order on such groups is defined in a natural way.Comment: 25 pages, 4 figures, LaTeX 2.0

Compactified jacobians

Let J be the jacobian of a reduced projective curve C with nodes only. 1) We give a simple and natural definition for its many compactifications and show the connection with various other definitions appearing in the literature. 2) Among all compactifications we choose one canonical, and define a theta divisor on it. 3) We give two very explicit and simple descriptions of a stratification of this canonical compactification into homogeneous spaces over J.Comment: AMSLaTeX 1.2/LaTeX2e with epic.sty, eepic.sty, Postscript file is also available at http://domovoy.math.uga.edu/preprints , a few misprints correcte