Geography of irreducible plane sextics

We complete the equisingular deformation classification of irreducible singular plane sextic curves. As a by-product, we also compute the fundamental groups of the complement of all but a few maximizing sextics

On the Artal-Carmona-Cogolludo construction

Cataloged from PDF version of article.We derive explicit defining equations for a number of irreducible maximizing plane sextics with double singular points only. For most real curves, we also compute the fundamental group of the complement; all groups found are abelian, which suffices to complete the computation of the groups of all non-maximizing irreducible sextics. As a by-product, examples of Zariski pairs in the strongest possible sense are constructed. © 2014 World Scientific Publishing Company

Transcendental lattice of an extremal elliptic surface

Cataloged from PDF version of article.We develop an algorithm computing the transcendental lattice and the Mordell-Weil group of an extremal elliptic surface. As an example, we compute the lattices of four exponentially large series of surfaces

The Alexander module of a trigonal curve

Cataloged from PDF version of article.We describe the Alexander modules and Alexander polynomials (both over ℚ and over finite fields Fp) of generalized trigonal curves. The rational case is completely resolved; in the case of characteristic p > 0, a few points remain open. The results obtained apply as well to plane curves with deep singularities. © European Mathematical Society.We describe the Alexander modules and Alexander polynomials (both over Q and over finite fields Fp) of generalized trigonal curves. The rational case is closed completely; in the case of characteristic p > 0, a few points remain open. The results obtained apply as well to plane curves with deep singularities

Real trigonal curves and real elliptic surfaces of type I

Cataloged from PDF version of article.We study real trigonal curves and elliptic surfaces of type I (over a base of an arbitrary genus) and their fiberwise equivariant deformations. The principal tool is a real version of Grothendieck's dessins d'enfants. We give a description of maximally inflected trigonal curves of type I in terms of the combinatorics of sufficiently simple graphs and, in the case of the rational base, obtain a complete classification of such curves. As a consequence, these results lead to conclusions concerning real Jacobian elliptic surfaces of type I with all singular fibers real. © De Gruyter 2014

Stable symmetries of plane sextics

We classify projective symmetries of irreducible plane sextics with simple singularities which are stable under equivariant deformations. We also outline a connection between order~2 stable symmetries and maximal trigonal curves

Conics in sextic K3K3-surfaces in P4\mathbb{P}^4

We prove that the maximal number of conics in a smooth sextic $K3$-surface $X\subset\mathbb{P}^4$ is 285, whereas the maximal number of real conics in a real sextic is 261. In both extremal configurations, all conics are irreducible