We give an explicit determinant formula for a class of rational solutions of the Painlevé V equation in terms of the universal characters.

Let $\Omega$ be a bounded pseudoconvex domain in ${\bf C}^n$. We give sufficient conditions for the Bergman metric to go to infinity uniformly at some boundary point, which is stated by the existence of a Hölder continuous plurisubharmonic peak function at this point or the verification of property $(P)$ (in the sense of Coman) which is characterized by the pluricomplex Green function.

In this paper we prove that a normal Gorenstein surface dominated by $\mathbf{P}2$ is isomorphic to a quotient $\mathbf{P}^2/G$, where $G$ is a finite group of automorphisms of $\mathbf{P}^2$ (except possibly for one surface $V_8'$). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.

The concepts of $\Lambda$-shellability of locally finite posets as well as of extendable sequentially Koszul algebras will be introduced. It will be proved that the divisor poset of a homogeneous semigroup ring is $\Lambda$-shellable if and only if the semigroup ring is extendable sequentially Koszul. Examples of extendable sequentially Koszul semigroup rings contain all monomial ASL's (algebras with straightening laws) and all second squarefree Veronese subrings.

In this paper we consider a subfield $K$ in a cyclotomic field $k_m$ of conductor $m$ such that $\left[k_m : K\right] = 2$ in the cases of $m = \ell p^n$ with a prime $p,$ where $\ell = 4$ or $p > \ell = 3.$ Then the theme is to know whether the ring of integers in $K$ has a power basis or does not.

Let $f : X \to B$ be a fiber space over a curve $B$ whose general fiber $F$ belongs to one of the following type: 1) $F$ is of general type and satisfying some mild conditions, 2) $F$ is with trivial canonical sheaf. In this note, a numerical characterization for $f : X \to B$ to be birationally trivial is given.

We prove the vanishing and non-vanishing theorems for an intersection of a finite number of $q$-complete domains in a complex manifold of dimension $n$. When $q$ does not divide $n$, it is stronger than the result naturally obtained by combining the approximation theorem of Diederich-Fornaess for $q$-convex functions with corners and the vanishing theorem of Andreotti-Grauert for $q$-complete domains. We also give an example which implies our result is best possible.

We show that the moduli space of abelian surfaces with polarisation of type $(1, 6)$ and a bilevel structure has positive Kodaira dimension and indeed $p_{g} \geq 3$. To do this we show that three of the Siegel cusp forms with character for the paramodular symplectic group constructed by Gritsenko and Nikulin are cusp forms without character for the modular group associated to this moduli problem. We then calculate the divisors of the corresponding differential forms, using information about the fixed loci of elements of the paramodular group previously obtained by Brasch.

Let $X$ be a smooth $n$-dimensional projective variety over an algebraically closed field $k$ such that $K_X$ is not nef. We give a characterization of non nef extremal rays of $X$ of maximal length (i.e of length $n-1$); in the case of Char$(k) = 0$ we also characterize non nef rays of length $n-2$.

We construct a Glauber dynamics on $\{ 0, 1 \}^{\mathcal{R}}$, $\mathcal{R}$ a discrete space, with infinite range flip rates, for which a fermion point process is reversible. We also discuss the ergodicity of the corresponding Markov process and the log-Sobolev inequality.