81 citations · 371 across the 49 of their papers we have counts for
46 papers · 1 filter
Lines on K3-quartics via triangular sets
Alex Degtyarev, Sławomir Rams
We prove the sharp upper bound of at most lines on a complex K3-surface of degree four with a non-empty singular locus. We also classify the configurations of more than l…
Conics on Barth--Bauer octics
Alex Degtyarev
We analyze the configurations of conics and lines on a special class of Kummer octic surfaces. In particular, we bound the number of conics by and show that there is a unique…
Conics in Kummer quartics
Alex Degtyarev
We classify the configurations of lines and conics in smooth Kummer quartics, assuming that all Kummer divisors map to conics. We show that the number of conics on such a quar…
Planes in cubic fourfolds
Alex Degtyarev, Ilia Itenberg, John Christian Ottem
We show that the maximal number of planes in a complex smooth cubic fourfold in is , realized by the Fermat cubic only; the maximal number of real planes in a…
Counting Lines with Vinberg's algorithm
Alex Degtyarev, Sławomir Rams
We combine classical Vinberg's algorithms with the lattice-theoretic/arithmetic approach from arXiv:1706.05734 [math.AG] to give a method of classifying large line configurations o…
800 conics in a smooth quartic surface
Alex Degtyarev
We construct an example of a smooth spatial quartic surface that contains 800 irreducible conics.
Conics in sextic -surfaces in
Alex Degtyarev
We prove that the maximal number of conics in a smooth sextic -surface is 285, whereas the maximal number of real conics in a real sextic is 261. In both…
Real algebraic curves with large finite number of real points
Erwan Brugallé, Alex Degtyarev, Ilia Itenberg +1
We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known…
Lines on smooth polarized -surfaces
Alex Degtyarev
For each integer , we give a sharp bound on the number of lines contained in a smooth complex -polarized -surface in . In the two most interesting…
Smooth models of singular -surfaces
Alex Degtyarev
We show that the classical Fermat quartic has exactly three smooth spatial models. As a generalization, we give a classification of smooth spatial (as well as some other) models of…
Lines in supersingular quartics
Alex Degtyarev
We show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if the characteristic of the field equals 2, and it is 112, 58, or at most 52, if…
Lines on quartic surfaces
Alex Degtyarev, Ilia Itenberg, Ali Sinan Sertöz
We show that the maximal number of (real) lines in a (real) nonsingular spatial quartic surface is 64 (respectively, 56). We also give a complete projective classification of all q…
Projective spaces in Fermat varieties
Alex Degtyarev
We give a brief systematic overview of a few results concerning the Néron--Severi lattices of Fermat varieties and Delsarte surfaces.
Geography of irreducible plane sextics
Ayşegül Akyol, Alex Degtyarev
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 al…
On the topology of projective subspaces in complex Fermat varieties
Alex Degtyarev, Ichiro Shimada
Let X be the complex Fermat variety of dimension n=2d and degree m>2. We investigate the submodule of the middle homology group of X with integer coefficients generated by the clas…
On the Picard group of a Delsarte surface
Alex Degtyarev
We suggest an algorithm computing, in some cases, an explicit generating set for the Néron--Severi lattice of a Delsarte surface.
Lines generate the Picard groups of certain Fermat surfaces
Alex Degtyarev
We answer a question of T.Shioda and show that, for any positive integer prime to 6, the Picard group of the Fermat surface is generated by the classes of lines contained…
On the Artal--Carmona--Cogolludo construction
Alex Degtyarev
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 g…
The Bertini involution
Alex Degtyarev
We summarize and extend E. Moody's results on the explicit equations related to the Bertini involution.
On plane sextics with double singular points
Alex Degtyarev
We compute the fundamental groups of five maximizing sextics with double singular points only; in four cases, the groups are as expected. The approach used would apply to other sex…
The Alexander module of a trigonal curve. II
Alex Degtyarev
We complete the enumeration of the possible roots of the Alexander polynomial (both conventional and over finite fields) of a trigonal curve. The curves are not assumed proper or i…
Products of pairs of Dehn twists and maximal real Lefschetz fibrations
Alex Degtyarev, Nermin Salepci
We address the problem of existence and uniqueness of a factorization of a given element of the modular group into a product of two Dehn twists. As a geometric application, we conc…
Real trigonal curves and real elliptic surfaces of type I
Alex Degtyarev, Ilia Itenberg, Victor Zvonilov
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 vers…
The Alexander module of a trigonal curve
Alex Degtyarev
We describe the Alexander modules and Alexander polynomials (both over $\Q$ and over finite fields $\FF{p}$) of generalized trigonal curves. The rational case is closed completely;…