Papers by Eugenii Shustin
arXiv (Cornell University), Oct 18, 2019
The family of complex projective surfaces in P 3 of degree d having precisely δ nodes as their on... more The family of complex projective surfaces in P 3 of degree d having precisely δ nodes as their only singularities has codimension δ in the linear system |O P 3 (d)| for sufficiently large d and is of degree N P 3 δ,C (d) = (4(d − 1) 3) δ /δ! + O(d 3δ−3). In particular, N P 3 δ,C (d) is polynomial in d. By means of tropical geometry, we explicitly describe (4d 3) δ /δ! + O(d 3δ−1) surfaces passing through a suitable generic configuration of n = d+3 3 − δ − 1 points in P 3. These surfaces are close to tropical limits which we characterize combinatorially, introducing the concept of floor plans for multinodal tropical surfaces. The concept of floor plans is similar to the well-known floor diagrams (a combinatorial tool for tropical curve counts): with it, we keep the combinatorial essentials of a multinodal tropical surface S which are sufficient to reconstruct S. In the real case, we estimate the range for possible numbers of real multi-nodal surfaces satisfying point conditions. We show that, for a special configuration w of real points, the number N P 3 δ,R (d, w) of real surfaces of degree d having δ real nodes and passing through w is bounded from below by 3 2 d 3 δ /δ! + O(d 3δ−1). We prove analogous statements for counts of multinodal surfaces in P
Mathematische Annalen, Jul 14, 2018
We show that every polynomially integrable planar outer convex billiard is elliptic. We also prov... more We show that every polynomially integrable planar outer convex billiard is elliptic. We also prove an extension of this statement to non-convex billiards. Contents
Summer School in Algebraic and Tropical Geometry
The mini-course was devoted to integral-valued invariants in real enumerative geometry. The main ... more The mini-course was devoted to integral-valued invariants in real enumerative geometry. The main example was the Welschinger invariant that counts real rational curves on real rational algebraic surfaces. The theory of Welschinger invariants has been discussed from various points of view: computation of some examples via topological methods, invariance of Welschinger count for the plane and other real rational surfaces, lack of invariance for higher genus curves, tropical approach including tropical Gromov-Witten, Welschinger and rened Block-Goettsche invariants and correspondence theorems, computational aspects, notably, Caporaso-Harris type recursive formulas and the main consequences - the positivity and logarithmic asymptotics of totally real Welschinger invariants of the plane.
International Mathematics Research Notices
Introduction Let D be a divisor on the smooth projective surface Sigma and denote by V = V jDj (S... more Introduction Let D be a divisor on the smooth projective surface Sigma and denote by V = V jDj (S 1 ; : : : ; S r ) the variety of irreducible curves C 2 jDj having exactly r singularities of (topological or analytic) types S 1 ; : : : ; S r . We say that V has the T-property at C 2 V , if the conditions imposed by the individual singularities of C are independent (or transversal), that is, if V is smooth of the expected codimension. For Sigma = P 2 it is well-known that the T-property holds for families of nodal curves (cf. [S]). But for more complicated singularities (beginning with cusps) there are examples, where the T-property fails ([Wa1, Lu, L
arXiv (Cornell University), Nov 9, 2016
The surfaces considered are real, rational and have a unique smooth real (−2)curve. Their canonic... more 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.
arXiv (Cornell University), Aug 22, 2006
We define a series of relative tropical Welschinger-type invariants of real toric surfaces. In th... more We define a series of relative tropical Welschinger-type invariants of real toric surfaces. In the Del Pezzo case, these invariants can be seen as real tropical analogs of relative Gromov-Witten invariants, and are subject to a recursive formula. As application we obtain new formulas for Welschinger invariants of real toric Del Pezzo surfaces.
arXiv (Cornell University), Feb 9, 2023
We show that the commutator relations in the refined tropical vertex group can be expressed via t... more We show that the commutator relations in the refined tropical vertex group can be expressed via the enumeration of suitable real rational curves in toric surfaces.
Journal of Algebraic Geometry, Nov 2, 2021
The family of complex projective surfaces in P 3 of degree d having precisely δ nodes as their on... more The family of complex projective surfaces in P 3 of degree d having precisely δ nodes as their only singularities has codimension δ in the linear system |O P 3 (d)| for sufficiently large d and is of degree N P 3 δ,C (d) = (4(d − 1) 3) δ /δ! + O(d 3δ−3). In particular, N P 3 δ,C (d) is polynomial in d. By means of tropical geometry, we explicitly describe (4d 3) δ /δ! + O(d 3δ−1) surfaces passing through a suitable generic configuration of n = d+3 3 − δ − 1 points in P 3. These surfaces are close to tropical limits which we characterize combinatorially, introducing the concept of floor plans for multinodal tropical surfaces. The concept of floor plans is similar to the well-known floor diagrams (a combinatorial tool for tropical curve counts): with it, we keep the combinatorial essentials of a multinodal tropical surface S which are sufficient to reconstruct S. In the real case, we estimate the range for possible numbers of real multi-nodal surfaces satisfying point conditions. We show that, for a special configuration w of real points, the number N P 3 δ,R (d, w) of real surfaces of degree d having δ real nodes and passing through w is bounded from below by 3 2 d 3 δ /δ! + O(d 3δ−1). We prove analogous statements for counts of multinodal surfaces in P
Tropical Geometry: new directions
Oberwolfach Reports, Jun 3, 2020
arXiv (Cornell University), May 20, 2019
In this note, we investigate the maximal number of intersection points of a line with the contour... more In this note, we investigate the maximal number of intersection points of a line with the contour of hypersurface amoebas in R n. We define the latter number to be the R-degree of the contour. We also investigate the R-degree of related sets such as the boundary of amoebas and the amoeba of the real part of hypersurfaces defined over R. For all these objects, we provide bounds for the respective R-degrees.
arXiv (Cornell University), Sep 13, 2022
Welschinger invariants (open Gromov-Witten invariants) enumerate real nodal rational curves in th... more Welschinger invariants (open Gromov-Witten invariants) enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of integral-valued enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, we show that through any generic configuration of four pairs of complex conjugate points, one always can trace a real rational three-cuspidal quartic.
arXiv (Cornell University), May 24, 2001
We introduce a class of combinatorial hypersurfaces in the complex projective space, i.e., subman... more We introduce a class of combinatorial hypersurfaces in the complex projective space, i.e., submanifolds of codimension 2 which are topologically "glued" out of affine algebraic hypersurfaces. Our construction can be viewed as a complex version of the Viro gluing theorem, relating topology of real algebraic hypersurfaces to the combinatorics of subdivisions of Newton polyhedra. If a subdivision is regular, the combinatorial hypersurface is isotopic to an algebraic hypersurface, if not, then the combinatorial hypersurface is an almost complex variety which possess many properties of genuine algebraic hypersurfaces, and in the real case, the real combinatorial hypersurfaces satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces.
arXiv (Cornell University), Jan 15, 2015
In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose... more In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.
arXiv (Cornell University), Aug 16, 2011
We give a recursive formula for purely real Welschinger invariants of real Del Pezzo surfaces of ... more We give a recursive formula for purely real Welschinger invariants of real Del Pezzo surfaces of degree K 2 ≥ 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.
Steady modes and sliding modes in the relay control systems with time delay
The relay control systems with time delay are considered. We show that the time delay does not al... more The relay control systems with time delay are considered. We show that the time delay does not allow to realize an ideal sliding mode, but implies oscillations, whose stability is determined by one discrete parameter-oscillation frequency. Any motion of scalar discontinuous delay system turns into a steady mode-a motion with a constant frequency. Such steady modes have all properties of
Oscillations in a second-order discontinuous system with delay
Discrete and Continuous Dynamical Systems, Dec 1, 2002
ABSTRACT The authors consider the equation αx '' (t)=-x ' (t)+F(x(t),t)-s... more ABSTRACT The authors consider the equation αx '' (t)=-x ' (t)+F(x(t),t)-signx(t-h), where F is a smooth function and α,h are positive constants. They study the dynamics of oscillations with emphasis on the existence, frequency and stability of periodic oscillations. The main conclusion of the paper is that, in the autonomous case F(x,t)≡F(x), for |F(x)|<1, there are periodic solutions with different frequencies of oscillations, though only slowly-oscillating solutions are (orbitally) stable. Also, the uniqueness of a periodic slowly-oscillating solution is shown under extra conditions, and a criterion for the existence of bounded oscillations in the case of unbounded function F(x,t) is given.
On the intersection of the close algebraic curves
Springer eBooks, 1984
Without Abstract
arXiv (Cornell University), Jul 26, 2016
We show that every polynomially integrable planar outer convex billiard is elliptic. We also prov... more We show that every polynomially integrable planar outer convex billiard is elliptic. We also prove an extension of this statement to non-convex billiards. Contents
Corrigendum to “A flexible affine 𝑀-sextic which is algebraically unrealizable”
Journal of Algebraic Geometry, Aug 28, 2019
We prove the algebraic unrealizability of a certain isotopy type of plane affine real algebraic M... more We prove the algebraic unrealizability of a certain isotopy type of plane affine real algebraic M M -sextic which is pseudoholomorphically realizable. This result completes the classification up to isotopy of real algebraic affine M M -sextics. The proof of this result given in a previous paper by the first two authors [J. Algebraic Geom. 11 (2002), pp. 293–310] was incorrect.
Crelle's Journal, Jan 30, 2002
We give an isotopy classification of real pseudo-holomorphic and real algebraic M-curves of degre... more We give an isotopy classification of real pseudo-holomorphic and real algebraic M-curves of degree 8 on the quadratic cone arranged in some special way with respect to a line, and show that there exist real pseudo-holomorphic curves which are not isotopic to any real algebraic curve in this class. In a similar way we find a pseudoholomorphic real plane a‰ne sextic which is not isotopic to a real algebraic sextic. The proofs are based on the braid group technique and highly singular degenerations of algebraic curves.. .. Ordnung angeht, so habe ich mich-freilich auf einem recht umständlichen Wege-davon ü berzeugt, daß die 11 Zü ge, die sie nach Harnack haben kann, keinesfalls sämtlich außerhalb von einander verlaufen dü rfen, sondern daß ein Zug existieren muß, in dessen Innerem ein Zug und in dessen Ä ußerem neun Zü ge verlaufen oder umgekehrt. D. Hilbert. ''Mathematische Probleme'' The authors were supported by a grant from the French-Israeli scientific cooperation program ''Arc-En-Ciel 2000'' (project no. 8).
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Papers by Eugenii Shustin