Tropical curves with a singularity in a fixed point (2009)
Markwig, Hannah, Markwig, Thomas, Shustin, Eugenii
In this paper, we study tropicalisations of families of curves with a singularity in a fixed point. The tropicalisation of such a family is a linear tropical variety. We describe its maximal...
Kaplan, Haim, Sharir, Micha, Shustin, Eugenii
Let $L$ be a set of $n$ lines in $\reals^d$, for $d\ge 3$. A {\em joint} of $L$ is a point incident to at least $d$ lines of $L$, not all in a common hyperplane. Using a very simple algebraic proof...
Tropical and algebraic curves with multiple points (2009)
Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core of the tropical enumerative geometry. We present a new version of a...
Shustin, Eugenii, Fridman, Leonid, Fridman, Emilia, Castaños, Fernando
We present sufficient conditions for robust relay-delayed semiglobal stabilization of second order systems, which relate the upper bound to an uncertain time delay and the parameters of the...
Shustin, Eugenii, Fridman, Leonid, Fridman, Emilia, Castaños, Fernando
We present sufficient conditions for robust relay-delayed semiglobal stabilization of second order systems, which relate the upper bound to an uncertain time delay and the parameters of the...
New cases of logarithmic equivalence of Welschinger and Gromov-Witten invariants (2007)
Itenberg, Ilia, Kharlamov, Viatcheslav, Shustin, Eugenii
We consider the product of two projective lines equipped with the complex conjugation transforming $(x,y)$ into $(\bar{y},\bar{x})$ and blown up in at most two real, or two complex conjugate, points....
A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces (2007)
Itenberg, Ilia, Kharlamov, Viatcheslav, Shustin, Eugenii
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...
New cases of logarithmic equivalence of Welschinger and Gromov-Witten invariants (2006)
Itenberg, Ilia, Kharlamov, Viatcheslav, Shustin, Eugenii
We consider the product of two projective lines equipped with the complex conjugation transforming $(x,y)$ into $(\bar{y},\bar{x})$ and blown up in at most two real, or two complex conjugate, points....
Equisingular Families of Projective Curves (2006)
Greuel, Gert-Martin, Lossen, Christoph, Shustin, Eugenii
In this survey, we report on progress concerning families of projective curves with fixed number and fixed (topological or analytic) types of singularities. We are, in particular, interested in...
Appendix to "Welschinger invariant and enumeration of real rational curves" (2006)
Itenberg, Ilia, Kharlamov, Viatcheslav, Shustin, Eugenii
The invariance of the Welschinger numbers for real unnodal Del Pezzo surfaces, which we used for the enumeration of real rational curves on real toric Del Pezzo surfaces (see math.AG/0303378 and IMRN...
A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces (2006)
Itenberg, Ilia, Kharlamov, Viatcheslav, Shustin, Eugenii
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...
An Improved Stability Method for Linear Systems with Fast-Varying Delays (2005)
Shustin, Eugenii, Fridman, Emilia
Stability of linear systems with uncertain bounded time-varying delays is studied under assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such...
A tropical Nullstellensatz (2005)
Shustin, Eugenii, Izhakian, Zur
We suggest a version of Nullstellensatz over the tropical semiring, the real numbers equipped with operations of maximum and addition.
Logarithmic asymptotics of the genus zero Gromov-Witten invariants of the blown up plane (2004)
Itenberg, Ilia, Kharlamov, Viatcheslav, Shustin, Eugenii
We study the growth of the genus zero Gromov-Witten invariants GW_{nD} of the projective plane P^2_k blown up at k points (where D is a class in the second homology group of P^2_k). We prove that,...
Appendix to "Welschinger invariant and enumeration of real rational curves" (2003)
Itenberg, Ilia, Kharlamov, Viatcheslav, Shustin, Eugenii
The invariance of the Welschinger numbers for real unnodal Del Pezzo surfaces, which we used for the enumeration of real rational curves on real toric Del Pezzo surfaces (see math.AG/0303378 and IMRN...
Welschinger invariant and enumeration of real rational curves (2003)
Itenberg, Ilia, Kharlamov, Viatcheslav, Shustin, Eugenii
Welschinger's invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using...
Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry (2002)
We prove a new patchworking theorem for singular algebraic curves, which states the following. Given a complex toric threefold $Y$ which fibers over ${\mathbb C}$ with a reduced reducible zero fiber...
Analytic order of singular and critical points (2002)
We deal with the following closely related problems: (i) For a germ of a reduced plane analytic curve, what is the minimal degree of an algebraic curve with a singular point analytically equivalent...
Viro theorem and topology of real and complex combinatorial hypersurfaces (2001)
Itenberg, Ilia, Shustin, Eugenii
We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension~2 in $\C P^n$ and are topologically "glued" out of algebraic hypersurfaces in...
The variety of plane curves with ordinary singularities is not irreducible (2001)
Gert-martin Greuel, Christoph Lossen, Eugenii Shustin
Complex plane algebraic curves, a classical object of algebraic geometry, still provide a number of challenging problems. Among them are questions about the geometry of families of curves with given...
The variety of plane curves with ordinary singularities is not irreducible (2001)
Greuel, Gert-Martin, Lossen, Christoph, Shustin, Eugenii
We construct an infinite series of equisingular families of projective plane curves with ordinary multiple points such that each family has irreducible components of different dimensions. This...
Plane curves of minimal degree with prescribed singularities (2000)
Greuel, Gert-Martin, Lossen, Christoph, Shustin, Eugenii
We prove that there exists a positive $alpha$ such thatfor any integer mbox{$dge 3$} and any topological typesmbox{$S_1,dots,S_n$} of plane curve singularities,...
Geometry of families of nodal curves on the blown-up projective plane (2000)
Greuel, Gert-Martin, Lossen, Christop, Shustin, Eugenii
Let P2r be the projective plane blown up at r generic points. Denote by E0; E1; : : : ; Er the strict transform of a generic straight line on P2 and the exceptional divisors of the blown-up points on...
Castelnuvo Function, Zero-dimensional Schemes and Singular Plane Curves (2000)
Greuel, Gert-Martin, Lossen, Christoph, Shustin, Eugenii
We study families V of curves in P2(C) of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the...
Geometry of Equisingular Families of Curves (2000)
Greuel, Gert-Martin, Shustin, Eugenii
Singular algebraic curves, their existence, deformation, families (from the local and global point of view) attract continuous attention of algebraic geometers since the last century. The aim of our...
E.: Castelnuovo function, zero-dimensional schemes and singular plane curves (2000)
Gert-martin Greuel, Christoph Lossen, Eugenii Shustin
(C) of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the expected dimension), respectively to...
Castelnuovo function, zero-dimensional schemes and singular plane curves (1999)
Greuel, Gert-Martin, Lossen, Christoph, Shustin, Eugenii
We study families V of curves in P^2 of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the...
Geometry of Equisingular Families of Curves (1999)
Gert-Martin Greuel, Eugenii Shustin
this article. 1 Existence of plane curves with given singularities
Witten deformation and polynomial differential forms (1998)
Farber, Michael, Shustin, Eugenii
As is well-known, the Witten deformation of the De Rham complex computes the De Rham cohomology. In this paper we study the Witten deformation on a noncompact manifold and restrict it to differential...
Singularity which has no M-smoothing. (1998)
Kharlamov, Viatcheslav, Orevkov, Stepan, Shustin, Eugenii
The Harnack bound on the number of real components of a plane real algebraic curve has a natural local version which states that the number of closed real components obtained by a perturbation of a...
Maximal smoothings of real plane curve singular points. (1998)
Kharlamov, Viatcheslav, Risler, Jean-Jacques, Shustin, Eugenii
The local Harnack inequality bounds from above the number of ovals which can appear in a small perturbation of a singular point. As is known, there are singular points for which this bound is not...
Maximal smoothings of real plane curve singular points. (1998)
Kharlamov, Viatcheslav, Risler, Jean-Jacques, Shustin, Eugenii
The local Harnack inequality bounds from above the number of ovals which can appear in a small perturbation of a singular point. As is known, there are singular points for which this bound is not...
Singularity which has no M-smoothing. (1998)
Kharlamov, Viatcheslav, Orevkov, Stepan, Shustin, Eugenii
The Harnack bound on the number of real components of a plane real algebraic curve has a natural local version which states that the number of closed real components obtained by a perturbation of a...
Plane curves of minimal degree with prescribed singularities. Inventiones 133 (1998)
Gert-martin Greuel, Christoph Lossen, Eugenii Shustin
Abstract. We prove that there exists a positive ff such that for any integer d 3 and any topological types S 1; : : : ; Sn of plane curve singularities, satisfying (S1) + \Delta \Delta \Delta + (Sn)...
New Asymptotics in the Geometry of Equisingular Families of Curves (1998)
Gert-Martin Greuel, Christoph Lossen, D Kaiserslautern, Eugenii Shustin
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...
Plane curves of minimal degree with prescribed singularities (1997)
Greuel, Gert-Martin, Lossen, Christoph, Shustin, Eugenii
We prove that there exists a>0 such that for any integer d>2 and any topological types S_1,...,S_n of plane curve singularities, satisfying $\mu(S_1)+...+\mu(S_n) \leq ad^2$, there exists a reduced...
Geometry Of Families Of Nodal Curves On The Blown-Up Projective Plane (1997)
Gert-Martin Greuel, Christoph Lossen, Eugenii Shustin
. Let P 2 r be the projective plane blown up at r generic points. Denote by E 0 ; E 1 ; : : : ; Er the strict transform of a generic straight line on P 2 and the exceptional divisors of the blown--up...
On imaginary plane curves and Spin quotients of complex surfaces by complex conjugation (1996)
Finashin, Sergey, Shustin, Eugenii
It is proven that for any topological or analytical types of isolated singular points of plane curves, there exists a non-real irreducible plane algebraic curve of degree $d$ which goes through $d^2$...
Geometry of families of nodal curves on rational surfaces (1996)
Greuel, Gert-Martin, Lossen, Christoph, Shustin, Eugenii
Let P^2_r be the projective plane blown up at r generic points. Denote by E_0,E_1,...,E_r the strict transform of a generic straight line on P^2 and the exceptional divisors of the blown-up points on...