Related papers: On surfaces with zero vanishing cycles

Irreducible symplectic varieties are higher-dimensional analogues of K3 surfaces. In this paper, we prove the finiteness of twists of irreducible symplectic varieties via a fixed finite field extension of characteristic $0$. The main…

In this paper, we extend the notion of Schwarz reflection principle for smooth minimal surfaces to the discrete analogues for minimal surfaces, and use it to create global examples of discrete minimal nets with high degree of symmetry.

Let $k$ be an infinite finitely generated field of characteristic $p>0$. Fix a separated scheme $X$ smooth, geometrically connected, and of finite type over $k$ and a smooth proper morphism $f:Y\rightarrow X$. The main result of this paper…

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to…

We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves D(X). To do this we find, for each such surface X, the set of surfaces Y for which there exists a…

We obtain a complete list of smooth projective threefolds over $\mathbb C$ for which the dimension of the space of vanishing cycles (in $H^2$ of the smooth hyperplane section) equals $2$. We also obtain a complete list of rank 2 very ample…

For integers $k$, we consider the affine cubic surface $V_{k}$ given by $M({\bf x})=x_{1}^2 + x_{2}^2 +x_{3}^2 -x_{1}x_{2}x_{3}=k$. We show that for almost all $k$ the Hasse Principle holds, namely that $V_{k}(\mathbb{Z})$ is non-empty if…

In this paper, by using a special Euler-Ramanujan identity and the idea of Wick rotation, we show that a one-parameter family of solutions to the zero mean curvature equation in Lorentz-Minkowski $3$-space $\mathbb E_1^3$, namely…

This paper is an announcement of the minimal model theory for log surfaces in all characteristics and contains some related results including a simplified proof of the Artin-Keel contraction theorem in the surface case.

A conjecture of Voisin states that two points on a smooth projective complex variety whose algebra of holomorphic forms is generated in degree 2 are rationally equivalent to each other if and only if their difference lies in the third step…

We classify Enriques surfaces with smooth K3 cover and finite automorphism group in arbitrary positive characteristic. The classification is the same as over the complex numbers except that some types are missing in small characteristics.…

Generic polyhedra are interesting mathematical objects to study in their own right. In this paper, we initialize a systematic study of two-dimensional generic polyhedra with an eye towards applications to low-dimensional topology,…

We propose a "Bloch type" conjecture for surfaces: if the cup product map in coherent cohomology is zero, then all intersections of homologically trivial divisors should be zero in the Chow group of zero-cycles. We prove this conjecture for…

In this paper, we establish a weak version of the Kodaira vanishing theorem for surfaces in positive characteristic. As an application, we obtain some fundamental theorems in the minimal model theory for klt surfaces.

We first study hyperplane sections of some singular schemes over a field. We prove a Bertini theorem for the log smoothness of generic hyperplane sections of a large class of log smooth schemes over a log point. We also give an abstract…

A conjecture of Kotschick predicts that a compact K\"ahler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in…

The generalization of Bertrand's theorem to abstract surfaces of revolution without "equators" is proved. We prove a criterion for the existence on such a surface of exactly two central potentials (up to an additive and a multiplicative…

We use Poonen's closed point sieve to prove two independent results. First, we show that the obvious obstruction to embedding a curve in a smooth surface is the only obstruction over a perfect field, by proving the finite field analogue of…

For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the…

We introduce four new elementary short proofs of the famous K\"onig's theorem which characterizes bipartite graphs by absence of odd cycles.