Eugenii Shustin

Tropical Algebraic Geometry

Preface.- 1. Introduction to tropical geometry - Images under the logarithm - Amoebas - Tropical curves.- 2. Patchworking of algebraic varieties - Toric geometry - Viros patchworking method - Patchworking of singular algebraic surfaces - Tropicalization in the enumeration of nodal curves.- 3. Applications of tropical geometry to enumerative geometry - Tropical hypersurfaces - Correspondence theorem - Welschinger invariants.- Bibliography.

Introduction to Singularities and Deformations

Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and technical sciences. This book presents the basic singularity theory of analytic spaces, including local deformation theory, and the theory of plane curve singularities. Plane curve singularities are a classical object of study, rich of ideas and applications, which still is in the center of current research and as such provides an ideal introduction to the general theory. Deformation theory is an important technique in many branches of contemporary algebraic geometry and complex analysis. This introductory text provides the general framework of the theory while still remaining concrete. In the first part of the book the authors develop the relevant techniques, including the Weierstras preparation theorem, the finite coherence theorem etc., and then treat isolated hypersurface singularities, notably the finite determinacy, classification of simple singularities and topological and analytic invariants. In local deformation theory, emphasis is laid on the issues of versality, obstructions, and equisingular deformations. The book moreover contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the mu-constant stratum which is based on deformations of the parameterization. Computational aspects of the theory are discussed as well. Three appendices, including basic facts from sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained. The material, which can be found partly in other books and partly in research articles, is presented from a unified point of view for the first time. It is given with complete proofs, new in many cases. The book thus can serve as source for special courses in singularity theory and local algebraic and analytic geometry.

Welschinger invariant and enumeration of real rational curves

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 Mikhalkin’s approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer d, there exists a real rational curve of degree d through any collection of 3d − 1 real points in the projective plane, and, moreover, asymptotically in the logarithmic scale at least one third of the complex plane rational curves through a generic point collection are real. We also obtain similar results for curves on other toric Del Pezzo surfaces.

PLANE CURVES OF MINIMAL DEGREE WITH PRESCRIBED SINGULARITIES

Abstract. We prove that there exists a positive α such that for any integer d≥3 and any topological types S1,…,Sn of plane curve singularities, satisfying there exists a reduced irreducible plane curve of degree d with exactly n singular points of types S1,…,Sn, respectively. This estimate is optimal with respect to the exponent of d. In particular, we prove that for any topological type S there exists an irreducible polynomial of degree having a singular point of type S.

A tropical approach to enumerative geometry

A detailed algebraic-geometric background is presented for the tropical approach to enumeration of singular curves on toric surfaces, which consists of reduc- ing the enumeration of algebraic curves to that of non-Archimedean amoebas, the images of algebraic curves by a real-valued non-Archimedean valuation. This idea was proposed by Kontsevich and recently realized by Mikhalkin, who enumerated the nodal curves on toric surfaces. The main technical tools are a refined tropicalization of one-parametric equisingular families of curves and the patchworking construction of singular algebraic curves. The case of curves with a cusp and the case of real nodal curves are also treated. §1. Introduction The rapid development of tropical algebraic geometry over recent years has led to interesting applications to enumerative geometry of singular algebraic curves, proposed by Kontsevich (16). The first result in this direction was obtained by Mikhalkin (18), who counted the curves with a given number of nodes on toric surfaces via lattice paths in con- vex lattice polygons. Our main goal in the present paper is to explain this breakthrough result, notably the link between algebraic curves and non-Archimedean amoebas, which is the core of the tropical approach to enumerative geometry. Our point of view is purely algebraic-geometric and differs from Mikhalkins method, which is based on symplectic geometry techniques. Briefly speaking, we count equisingular families of curves over a punctured disk. The tropicalization procedure extends such families to the central point, and these tropical limits are basically encoded by non-Archimedean amoebas. In its turn, the patchworking construction restores an equisingular family out of the central fiber. Tropicalization. Let ∆ ⊂ R 2 be a convex lattice polygon, and let TorK(∆) be the toric surface associated with the polygon ∆ and defined over an algebraically closed field K of characteristic zero. We denote by ΛK(∆) the tautological linear system on TorK(∆) generated by the monomials x i y j ,( i, j) ∈ ∆ ∩ Z 2 . We would like to count the n-nodal curves belonging to ΛK(∆) and passing through r =d im Λ K(∆) − n = |∆ ∩ Z 2 |− 1 − n generic points in TorK(∆), i.e., we want to find the degree of the so-called Severi variety Σ∆(nA1). Let K be the field of convergent Puiseux series over C, i.e., power series of the form b(t )= � τ ∈R cτ t τ ,w hereR ⊂ Q is contained in an arithmetic progression bounded from below, and � τ ∈R |cτ |t τ < ∞ for sufficiently small positive t. The field K is equipped with a non-Archimedean valuation Val(b )= − min{τ ∈ R : cτ � } ,w hich

Gluing of singular and critical points

Abstract We suggest an approach to construction of algebraic hypersurfaces with given collection of singular points and polynomials with given collection of critical points. The approach is based on the Viro method of gluing polynomials and on the geometry of equisingular and equicritical strata in spaces of polynomials. As application we construct cuspidal plane curves of small degrees and real polynomials in two variables with given numbers of degenerate and non-degenerate critical points.

Brief paper: On delay-derivative-dependent stability of systems with fast-varying delays

Stability of linear systems with uncertain bounded time-varying delays is studied under the assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound of the L2-norm of a certain integral operator. There exists a bound on this operator norm in two cases: in the case where the delay derivative is not greater than 1 and in the case without any constraints on the delay derivative. In the present note we fill the gap between the two cases by deriving a tight operator bound which is an increasing and continuous function of the delay derivative upper bound d≥. For d➝∞ the new bound corresponds to the second case and improves the existing bound. As a result, for the first time, delay-derivative-dependent frequency domain and time domain stability criteria are derived for systems with the delay derivative greater than 1.

A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces

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.

Logarithmic equivalence of Welschinger and Gromov-Witten invariants

The Welschinger numbers, a kind of a real analogue of the Gromov-Witten numbers that count the complex rational curves through a given generic collection of points, bound from below the number of real rational curves for any generic collection of real points. Logarithmic equivalence of sequences is understood to mean the asymptotic equivalence of their logarithms. Such an equivalence is proved for the Welschinger and Gromov-Witten numbers of any toric Del Pezzo surface with its tautological real structure, in particular, of the projective plane, under the hypothesis that all, or almost all, the chosen points are real. A study is also made of the positivity of Welschinger numbers and their monotonicity with respect to the number of imaginary points.

Real plane algebraic curves with prescribed singularities

IN THIS article we deal with the classical problem about a number of prescribed singularities on a plane algebraic curve of a given degree. We present constructions of real plane algebraic curves of a given degree with arbitrary singularities. In particular, we get irreducible curves of degree d with any number of real cusps between 0 and d2/4 + O(d). It is well-known [9] that for any positive integers d, m satisfying