S. Kondō defined a birational period map between the moduli space of genus three curves and a moduli space of polarized K3 surfaces. In this paper we give a resolution of the period map, providing a surjective morphism from a suitable compactification of $\mathcal{M}_{3}$ to the Baily-Borel compactification of a six dimensional ball quotient.

It is well-known that the moduli space $\overline{M}_{g}$ of Deligne-Mumford stable curves of genus $g$ admits a stratification by the loci of stable curves with a fixed number $i$ of nodes, where $0 \le i \le 3g-3$. There is an analogous stratification of the associated moduli stack $\overline{\mathcal{M}}_{g}$.

In this paper we are interested in that particular stratum of the moduli stack, which corresponds to stable curves with exactly $3g-4$ nodes. The irreducible components of this stratum are one-dimensional substacks of $\overline{\mathcal{M}}_{g}$. We show how these substacks can be related to simpler moduli stacks of (permutation classes of) pointed stable curves. Furthermore, we use this to construct all of the components of this boundary stratum generically in a new way as explicit quotient stacks.

Bounds for the maximum degree of a minimal Gröbner basis of simplicial toric ideals with respect to the reverse lexicographic order are given. These bounds are close to the bound stated in Eisenbud-Goto's Conjecture on the Castelnuovo-Mumford regularity.

A toric face ring, which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Römer and their coauthors recently. In this paper, under the "normality" assumption, we describe a dualizing complex of a toric face ring $R$ in a very concise way. Since $R$ is not a graded ring in general, the proof is not straightforward. We also develop the squarefree module theory over $R$, and show that the Cohen-Macaulay, Buchsbaum, and Gorenstein* properties of $R$ are topological properties of its associated cell complex.

We prove that the averaging formula for Nielsen numbers holds for continuous maps on infra-solvmanifolds of type (R): Let $M$ be such a manifold with holonomy group $\Psi$ and let $f:M \to M$ be a continuous map. The averaging formula for Nielsen numbers

N(f) = \frac{1}{\lvert\Psi\rvert} \sum_{A \in \Psi} \lvert\det(A_{*}-f_{*})\rvert

is proved. This is a workable formula for the difficult number $N(f)$.

In this paper, the characterization of domains in $\mathbb{C}^{n}$ by their noncompact automorphism groups are given.