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Ilia Itenberg

Publications and source records attributed to Ilia Itenberg.

At least 19 recordsLinked to original sources
arXiv2026

Quantum index, Arnold-Rokhlin surfaces, and real enumerative geometry

The main goal of this note is to relate two different Welschinger-type rules of signs that were used for definition of real enumerative invariants of toric surfaces (the invariants considered being relative to the toric boundary). The relation between them intertwines Mikhalkin's quantum index and geometry of real and complex point sets of counted real curves. As a by-product, we suggest a new definition of quantum index, which can be used for refined invariant enumeration of real algebraic curves on surfaces in a non-toric setup.

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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).

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arXiv2023

Real enumerative invariants relative to the anti-canonical divisor and their refinement

We introduce new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from the appropriate enumeration of real elliptic curves. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case. We also construct tropical counterparts of the refined elliptic invariants under consideration and establish a tropical algorithm allowing one to compute, {\it via} a suitable version of the correspondence theorem, the above invariants.

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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.

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arXiv2021

Area in real K3-surfaces

For a real K3-surface $X$, one can introduce areas of connected components of the real point set $\mathbb{R} X$ of $X$ using a holomorphic symplectic form of $X$. These areas are defined up to simultaneous multiplication by a positive real number, so the areas of different components can be compared. In particular, it turns out that the area of a non-spherical component of $\mathbb{R} X$ is always greater than the area of any spherical component. In this paper we explore further comparative restrictions on the area for real K3-surfaces admitting a suitable polarization of degree $2g - 2$ (where $g$ is a positive integer) and such that $\mathbb{R} X$ has one non-spherical component and at least $g$ spherical components. For this purpose we introduce and study the notion of simple Harnack curves in real K3-surfaces, generalizing planar simple Harnack curves.

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arXiv2020

Tropical Homology

Given a tropical variety X and two non-negative integers p and q we define homology group $H_{p,q}(X)$. We show that if X is a smooth tropical variety that can be represented as the tropical limit of a 1-parameter family of complex projective varieties, then $\dim H_{p,q}(X)$ coincides with the Hodge number $h^{p,q}$ of a general member of the family.

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arXiv2018

Hurwitz numbers for real polynomials

We consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points. We show that, provided the polynomials are counted with an appropriate sign, their number does not depend on the order of the branch points on the real line. We study generating series for the invariants thus obtained, determine necessary and sufficient conditions for the vanishing and nonvanishing of these generating series, and obtain a logarithmic asymptotic for the invariants as the degree of the polynomials tends to infinity.

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arXiv2018

Real algebraic curves with large finite number of real points

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 upper and lower bounds and construct close to optimal curves of small degree. Our upper bound is sharp if the genus is small as compared to the degree. Some of the results are extended to other real algebraic surfaces, most notably ruled.

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arXiv2018

Relative enumerative invariants of real nodal del Pezzo surfaces

The surfaces considered are real, rational and have a unique smooth real $(-2)$-curve. Their canonical class $K$ is strictly negative on any other irreducible curve in the surface and $K^2>0$. For surfaces satisfying these assumptions, we suggest a certain signed count of real rational curves that belong to a given divisor class and are simply tangent to the $(-2)$-curve at each intersection point. We prove that this count provides a number which depends neither on the point constraints nor on deformation of the surface preserving the real structure and the $(-2)$-curve.

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arXiv2018

Welschinger invariants of real del Pezzo surfaces of degree $\ge2$

We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree at least 2. We show that under some conditions, for such a surface $X$ and a real nef and big divisor class $D$, through any generic collection of $-DK_X-1$ real points lying on a connected component of the real part of $X$ one can trace a real rational curve $C\in|D|$. This is derived from the positivity of appropriate Welschinger invariants. We furthermore show that these invariants are asymptotically equivalent, in the logarithmic scale, to the corresponding genus zero Gromov-Witten invariants. Our approach consists in a conversion of Shoval-Shustin recursive formulas counting complex curves on the plane blown up at seven points and of Vakil's extension of the Abramovich-Bertram formula for Gromov-Witten nvariants into formulas computing real enumerative invariants.

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arXiv2017

Welschinger invariants revisited

We establish the enumerativity of (original and modified) Welschinger invariants for every real divisor on any real algebraic Del Pezzo surface and give an algebro-geometric proof of the invariance of that count both up to variation of the point constraints on a given surface and variation of the complex structure of the surface itself.

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arXiv2016

Rational quintics in the real plane

From a topological viewpoint, a rational curve in the real projective plane is generically a smoothly immersed circle and a finite collection of isolated points. We give an isotopy classification of generic rational quintics in $\mathbb{RP}^2$ in the spirit of Hilbert's 16th problem.

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arXiv2016

Lines on quartic surfaces

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 quartics containing more than 52 lines: all such quartics are projectively rigid. Any value not exceeding 52 can appear as the number of lines of an appropriate quartic.

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arXiv2015

Brief introduction to tropical geometry

The paper consists of lecture notes for a mini-course given by the authors at the Gökova Geometry \& Topology conference in May 2014. We start the exposition with tropical curves in the plane and their applications to problems in classical enumerative geometry, and continue with a look at more general tropical varieties and their homology theories.

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arXiv2015

Welschinger invariants of real Del Pezzo surfaces of degree $\ge 3$

We give a recursive formula for purely real Welschinger invariants of real Del Pezzo surfaces of degree $K^2\ge 3$, where in the case of surfaces of degree $3$ with two real components we introduce a certain modification of Welschinger invariants and enumerate exclusively the curves traced on the non-orientable component. As an application, we prove the positivity of the invariants under consideration and their logarithmic asymptotic equivalence, as well as congruence modulo $4$, to genus zero Gromov-Witten invariants.

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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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arXiv2012

On Block-Goettsche multiplicities for planar tropical curves

We prove invariance for the number of planar tropical curves enhanced with polynomial multiplicities recently proposed by Florian Block and Lothar Goettsche. This invariance has a number of implications in tropical enumerative geometry.

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