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Alex Degtyarev

Publications and source records attributed to Alex Degtyarev.

At least 19 recordsLinked to original sources
arXiv2025

Lines on K3-sextics with simple singularities

We advance our understanding of the configurations of low degree smooth rational curves on (quasi-)polarized complex K3-surfaces. We apply our efficient approach to classify the configurations of at least 36 lines on K3-sextics with at worst A-D-E singularities. Besides, we characterize a certain class of infinite dihedral groups of birational automorphisms of K3-sextics, we show that no K3-sextic can contain a Kummer configuration of lines, and we give a complete account of the line configurations on closest analogue of Kummer K3-octics or quartics, viz. the so-called Humbert K3-sextics.

math.AG↗
arXiv2025

Periodic sharpness of Miyaoka's bound for smooth rational curves

We determine the maximal number of smooth rational degree d curves on a complex K3-surface of degree 2n provided n is sufficiently large as compared to d>1. We obtain precise characterization of configurations of rational degree d curves for which Miyaoka's bound is sharp for nd odd and n sufficiently large as compared to d.

math.AG↗
arXiv2025

Split hyperplane sections on smooth polarized $K3$-surfaces

We study hyperplane sections of smooth polarized $K3$-surfaces that split into unions of lines. We describe the dual adjacency graphs of such sections and find sharp upper bounds on their number. In most cases (starting from degree $6$), we also compute all configurations of such sections.

math.AG↗
arXiv2025

On large configurations of lines on quartic surfaces

We estimate the number of lines on a non-K3 quartic surface. Such a surface with only isolated double point(s) contains at most twenty lines; this bound is attained by a unique configuration of lines and by a surface with a certain limited set of singularities. We have similar itemized bounds for other types of non-simple singularities, which culminate in at most 31 lines on a non-K3 quartic not ruled by lines; this bound is only attained on the quartic monoids described by K.~Rohn.

math.AG↗
arXiv2025

K3 surfaces of degree six arising from desmic tetrahedra

We study K3 surfaces of degree 6 containing two sets of 12 skew lines such that each line from a set intersects exactly six lines from the other set. These surfaces arise as hyperplane sections of the cubic line complex associated with the pencil of desmic quartic surfaces introduced by George Humbert and recently studied by the second and third authors. We discuss alternative birational models of the surfaces, compute the Picard lattice and a group of projective automorphisms, and describe rational curves of low degree on the general surface.

math.AG↗
arXiv2024

Conics on smooth quartic surfaces

We prove that the maximal number of conics, a priori irreducible of reducible, on a smooth spatial quartic surface is 800, realized by a unique quartic. We also classify quartics with many (at least 720) conics. The maximal number of real conics on a real quartics is between 656 and 718.

math.AG↗
arXiv2024

Real plane sextics without real points

We prove that the equisingular deformation type of a simple real plane sextic curve with smooth real part is determined by its real homological type, \ie, the polarization, exceptional divisors, and real structure recorded in the homology of the covering $K3$-surface. As an illustration, we obtain an equisingular deformation classification of real plane sextics with empty real part (for completeness, we consider the few non-simple ones as well).

math.AG↗
arXiv2023

Conics on Barth--Bauer octics

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 $176$ and show that there is a unique surface with $176$ conics, all irreducible: it admits a faithful action of one of the Mukai groups. Therefore, we also discuss conics and lines on Mukai surfaces: we discover a double plane (ramified at a smooth sextic curve) that contains $8910$ smooth conics.

math.AG↗
arXiv2023

Lines on K3-quartics via triangular sets

We prove the sharp upper bound of at most $52$ lines on a complex K3-surface of degree four with a non-empty singular locus. We also classify the configurations of more than $48$ lines on smooth complex quartics.

math.AG↗
arXiv2022

Planes in cubic fourfolds

We show that the maximal number of planes in a complex smooth cubic fourfold in ${\mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the so-called Clebsch--Segre cubic. Altogether, there are but three (up to projective equivalence) cubics with more than $350$ planes.

math.AG↗
arXiv2022

Slope and concordance of links

The slope is an isotopy invariant of colored links with a distinguished component, initially introduced by the authors to describe an extra correction term in the computation of the signature of the splice. It appeared to be closely related to several classical invariants, such as the Conway potential function or the Kojima eta-function (defined for two-components links). In this paper, we prove that the slope is invariant under colored concordance of links. Besides, we present a formula to compute the slope in terms of C-complexes and generalized Seifert forms.

math.GT↗
arXiv2021

Conics in Kummer quartics

We classify the configurations of lines and conics in smooth Kummer quartics, assuming that all $16$ Kummer divisors map to conics. We show that the number of conics on such a quartic is at most $800$.

math.AG↗
arXiv2021

Tritangents to smooth sextic curves

We prove that a smooth plane sextic curve can have at most 72 tritangents, whereas a smooth real sextic may have at most 66 real tritangents.

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arXiv2021

Conics in sextic $K3$-surfaces in $\mathbb{P}^4$

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.

math.AG↗
arXiv2021

Counting Lines with Vinberg's algorithm

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 on complex quasi-polarized K3-surfaces. We apply our method to classify all complex K3-octic surfaces with at worst Du Val singularities and at least 32 lines. The upper bound on the number of lines is 36, as in the smooth case, with at most 32 lines if the singular locus is non-empty.

math.AG↗
arXiv2021

Lines in supersingular quartics

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 the characteristic equals 3. If the quartic is not supersingular, the number of lines is at most 60 in both cases. We also give a complete classification of large configurations of lines.

math.AG↗