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Alex Degtyarev

Publications and source records attributed to Alex Degtyarev.

At least 55 records · Page 3Linked to original sources
arXiv2012

On the number of components of a complete intersection of real quadrics

Our main results concern complete intersections of three real quadrics. We prove that the maximal number $B^0_2(N)$ of connected components that a regular complete intersection of three real quadrics in $\Bbb{P}^N$ can have differs at most by one from the maximal number of ovals of the submaximal depth $[(N-1)/2]$ of a real plane projective curve of degree $d=N+1$. As a consequence, we obtain a lower bound \smash{$\frac14 N^2+O(N)$} and an upper bound \smash{$\frac38 N^2+O(N)$} for $B^0_2(N)$.

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arXiv2011

Real trigonal curves and real elliptic surfaces of type I

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 \emph{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.

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arXiv2010

On real determinantal quartics

We describe all possible arrangements of the ten nodes of a generic real determinantal quartic surface in $\Cp3$ with nonempty spectrahedral region.

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arXiv2010

Dihedral coverings of trigonal curves

We classify and study trigonal curves in Hirzebruch surfaces admitting dihedral Galois coverings. As a consequence, we obtain certain restrictions on the fundamental group of a plane curve~$D$ with a singular point of multiplicity $(°D-3)$.

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arXiv2009

Hurwitz equivalence of braid monodromies and extremal elliptic surfaces

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $Γ$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\it topologically} distinct algebraic objects such as extremal elliptic surfaces, real trigonal curves, and real elliptic surfaces.

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arXiv2009

The fundamental group of a generalized trigonal curve

We develop a modification of the Zariski--van Kampen approach for the computation of the fundamental group of a trigonal curve with improper fibers. As an application, we list the deformation families and compute the fundamental groups of all irreducible maximizing simple sextics with a type $\bold D$ singular point.

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arXiv2009

Plane sextics via dessins d'enfants

We develop a geometric approach to the study of plane sextics with a triple singular point. As an application, we give an explicit geometric description of all irreducible maximal sextics with a type $\bold E_7$ singular point and compute their fundamental groups. All groups found are finite; one of them is nonabelian.

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arXiv2009

Classical Zariski pairs

We compute the fundamental groups of all irreducible plane sextics constituting classical Zariski pairs

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arXiv2009

Plane sextics with a type $\bold E_8$ singular point

We construct explicit geometric models for and compute the fundamental groups of all plane sextics with simple singularities only and with at least one type $\bold E_8$ singular point. In particular, we discover four new sextics with nonabelian fundamental groups; two of them are irreducible. The groups of the two irreducible sextics found are finite. The principal tool used is the reduction to trigonal curves and Grothendieck's {\it dessins d'enfants}

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arXiv2008

Irreducible plane sextics with large fundamental groups

We compute the fundamental group of the complement of each irreducible sextic of weight eight or nine (in a sense, the largest groups for irreducible sextics), as well as of 169 of their derivatives (both of and not of torus type). We also give a detailed geometric description of sextics of weight eight and nine and of their moduli spaces and compute their Alexander modules; the latter are shown to be free over an appropriate ring.

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arXiv2008

On deformation types of real elliptic surfaces

We study real elliptic surfaces and trigonal curves (over a base of an arbitrary genus) and their equivariant deformations. We calculate the real Tate-Shafarevich group and reduce the deformation classification to the combinatorics of a real version of Grothendieck's {\it dessins d'enfants}. As a consequence, we obtain an explicit description of the deformation classes of $M$- and $(M-1)$- (i.e., maximal and submaximal in the sense of the Smith inequality) curves and surfaces.

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arXiv2008

Fundamental groups of symmetric sextics

We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with at least two type $\bold{E}_6$ singular points. As a simple application, we compute the fundamental groups of 125 other sextics, most of which are new.

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arXiv2008

Towards the generalized Shapiro and Shapiro conjecture

We find a new, asymptotically better, bound $g\le\frac14d^2+O(d)$ on the genus of a curve that may violate the generalized total reality conjecture. The bound covers all known cases except $g=0$ (the original conjecture).

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arXiv2008

Fundamental groups of symmetric sextics. II

We study the moduli spaces and compute the fundamental groups of plane sextics of torus type with the set of inner singularities $2\bold{A}_8$ or $\bold{A}_{17}$. We also compute the fundamental groups of a number of other sextics, both of and not of torus type. The groups found are simplest possible, i.e., $\Bbb{Z}_2*\Bbb{Z}_3$ and $\Bbb{Z}_6$, respectively.

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arXiv2008

Zariski $k$-plets via dessins d'enfants

We construct exponentially large collections of pairwise distinct equisingular deformation families of irreducible plane curves sharing the same sets of singularities. The fundamental groups of all curves constructed are abelian.

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