Related papers: Real Quartic Surfaces in RP_3 containing 16 Skew L…

The Hilbert scheme of projective 3-folds of codimension 3 or more that are linear scrolls over the projective plane or over a smooth quadric surface or that are quadric or cubic fibrations over the projective line is studied. All known such…

We prove new cases of the Hasse principle for Kummer surfaces constructed from 2-coverings of Jacobians of genus 2 curves, assuming finiteness of relevant Tate--Shafarevich groups. Under the same assumption, we deduce the Hasse principle…

Let $C/K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model…

We introduce enumerative invariants of real del Pezzo surfaces that count real rational curves belonging to a given divisor class, passing through a generic conjugation-invariant configuration of points and satisfying preassigned tangency…

We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K-theoretic invariants are…

Let $k$ be an algebraically closed field and ${\sf G}(2,k^4)$ the Grassmannian of 2-planes in $k^4$. We associate to each 6-dimensional subspace $R$ of the space of 4x4 matrices over $k$ a closed subscheme ${\bf X}_R \subseteq {\sf…

We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We…

Two Magma functions are given: one computes linear systems of plane curves with non-ordinary singularities and the other computes a scheme which parametrizes given degree plane curves with given singularities. These functions provide an…

We formulate an equivariant conservation of number, which proves that a generalized Euler number of a complex equivariant vector bundle can be computed as a sum of local indices of an arbitrary section. This involves an expansion of the…

In this paper we give a general construction of transcendental lattices for K3 surfaces with real multiplication by arbitrary field up to degree 6 along with formula for their discriminants. We also show that all simple Abelian fourfolds…

Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that is employed in order to quantify the possibility of…

In this paper, we study the classical theory of quadratic line complexes and Kummer surfaces. A quadratic line complex is the intersection of the Grassmannian $G(2,4)$ and a quadric hypersurface in ${\bf P}^5$, and a Kummer surface is the…

We present the various constructions of new symplectic $4$-manifolds with non-negative signatures using the complex surfaces on the BMY line $c_1^2 = 9\chi_h$, the Cartwright-Steger surfaces, the quotients of Hirzebruch's certain…

Let $X$ be a real algebraic variety with set of complex points $X_{\mathbb C}$ and set of real points $X_{\mathbb R}$. A complex slice of $X$ is a transverse intersection of $X_{\mathbb R}$ with a complex subvariety $V$ of $X_{\mathbb C}$.…

David Hilbert proved that a non-negative real quartic form f(x,y,z) is the sum of three squares of quadratic forms. We give a new proof which shows that if the complex plane curve Q defined by f is smooth, then f has exactly 8 such…

So far only six families of smooth irregular surfaces are known to exist in P^4 (up to pullbacks by suitable finite covers of P^4). These are the elliptic quintic scrolls, the minimal abelian and bielliptic surfaces (of degree 10), two…

This paper develops a discrete theory of real Riemann surfaces based on quadrilateral cellular decompositions (quad-graphs) and a linear discretization of the Cauchy-Riemann equations. We construct a discrete analogue of an antiholomorphic…

We give a new construction of the irregular, generalized Lagrangian, surfaces of general type with p_g=5, \chi=2, K^2=8, recently discovered by Chad Schoen. Our approach proves that, if S is a general Schoen surface, its canonical map is a…

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils…

In this paper, we give a Zariski triple of the arrangements for a smooth quartic and its four bitangents. A key criterion to distinguish the topology of such curves is given by a matrix related to the height pairing of rational points…