Broccoli curves and the tropical invariance of Welschinger numbers

In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points — for arbitrary choices of the directions of the ends of the curves, possibly with higher weights, and also if some of the ends are fixed. In the toric Del Pezzo case we show that these broccoli invariants are equal to the Welschinger invariants (with real and complex conjugate point conditions), thus providing a proof of the independence of Welschinger invariants of the point conditions within tropical geometry. The general case gives rise to a tropical Caporaso–Harris formula for broccoli curves which suffices to compute all Welschinger invariants of the plane.     

Welschinger invariants of real Del Pezzo surfaces of degree ≥ 3

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.     

A boundedness result for toric log Del Pezzo surfaces

In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index ℓ. This upper bound turns out to be a quadratic polynomial in the variable ℓ. Received: 18 June 2008     

Floor diagrams relative to a conic,and GW–W invariants of Del Pezzo surfaces

We enumerate, via floor diagrams, complex and real curves in CP²$\mathbb{C}{P}^2$ blown up in n points on a conic. As an application, we deduce Gromov–Witten and Welschinger invariants of Del Pezzo surfaces. These results are mainly obtained using Li's degeneration formula and its real counterpart.     

Kac–Moody $${\widetilde{E}_k}$$-bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Given an elliptic curve Σ, flat E _k -bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an A _d -singularity containing Σ as an anti-canonical curve and Kac–Moody [(E)\tilde]_k{\widetilde{E}_{k}}-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding [(E)\tilde]_k{\widetilde{E}_k}-bundles over Σ can be reduced to E _k -bundles.     

New cases of logarithmic equivalence of Welschinger and Gromov-Witten invariants

We consider ℙ¹ × ℙ¹ equipped with the complex conjugation (x, y) ↦ and blown up in at most two real or two complex conjugate points. For these four surfaces we prove the logarithmic equivalence of Welschinger and Gromov-Witten invariants. To Vladimir Igorevich Arnold on his 70th birthday     

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X _k obtained by blowing up ℂℙ² at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W _k :M _k →ℂ with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X _k , and give an explicit correspondence between the deformation parameters for X _k and the cohomology class [B+iω]∈H ²(M _k ,ℂ).     

On infinite real trace rational languages of maximum topological complexity

We consider the set ℝ^ω(Γ, D) of infinite real traces, over a dependence alphabet (Γ,D) with no isolated letter, equipped with the topology induced by the prefix metric. We prove that all rational languages of infinite real traces are analytic sets. We also reprove that there exist some rational languages of infinite real traces that are analytic but non-Borel sets; in fact, these sets are even Σ ₁ ¹ -complete, hence have maximum possible topological complexity. For this purpose, we give an example of a Σ ₁ ¹ -complete language that is fundamentally different from the known example of a Σ ₁ ¹ -complete infinitary rational relation given by Finkel (2003). Bibliography: 35 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 205–223.     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000     

Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry

Following the approach of Gromov and Witten, we construct invariants under deformation of real rational symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves in a given homology class passing through a given real configuration of points. To cite this article: J.-Y. Welschinger, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series

Abstract

The Hilbert series of a polarized algebraic variety (X, D) is a powerful invariant that, while it captures some features of the geometry of (X, D) precisely, often cannot recover much information about its singular locus. This work explores the extent to which the Hilbert series of an orbifold del Pezzo surface fails to pin down its singular locus, which provides nonexistence results describing when there are no orbifold del Pezzo surfaces with a given Hilbert series, supplies bounds on the number of singularities on such surfaces, and has applications to the combinatorics of lattice polytopes in the toric case.     

Rank 2 Ample Vector Bundles on Some Smooth Rational Surfaces

Several classification results for ample vector bundles of rank 2 on Hirzebruch surfaces, and on Del Pezzo surfaces, are obtained. In particular, we classify rank 2 ample vector bundles with _c2 less than seven on Hirzebruch surfaces, and with _c2 less than four on Del Pezzo surfaces.     

Bifurcation method for solving multiple positive solutions to Henon equation

Three algorithms based on the bifurcation method are applied to solving the D4 symmetric positive solutions to the boundary value problem of Henon equation. Taking r in Henon equation as a bi- furcation parameter, the D4-Σd(D4-Σ1, D4-Σ2) symmetry-breaking bifurcation points on the branch of the D4 symmetric positive solutions are found via the extended systems. Finally, Σd(Σ1, Σ2) sym- metric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction.     

Curve counting via stable pairs in the derived category

For a nonsingular projective 3-fold X, we define integer invariants virtually enumerating pairs (C,D) where C⊂X is an embedded curve and D⊂C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of X. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described.     

Kawamata–Viehweg vanishing on rational surfaces in positive characteristic

We prove that the Kawamata–Viehweg vanishing theorem holds on rational surfaces in positive characteristic by means of the lifting property to W ₂(k) of certain log pairs on smooth rational surfaces. As a corollary, the Kawamata–Viehweg vanishing theorem holds on log del Pezzo surfaces in positive characteristic.     

The integral cohomology of toric manifolds

We prove that the integral cohomology of a smooth, not necessarily compact, toric variety X _Σ is determined by the Stanley-Reisner ring of Σ. This follows from a formality result for singular cochains on the Borel construction of X _Σ. As a consequence, we show that the cycle map from Chow groups to Borel-Moore homology is split injective.     

Abelian surfaces in projective toric 4-folds

We show fundamental properties on embedding of abelian surfaces into projective toric 4-folds, and study the case of the toric Del Pezzo 4-fold from the viewpoint of the moduli space of abelian surfaces with polarization of type (1, 5). Received: 31 January 2005; revised: 15 April 2005     

Branched immersions and braids

Branch points of a real 2-surface Σ in a 4-manifold M generalize branch points of complex curves in complex surfaces: for example, they can occur as singularities of minimal surfaces. We investigate such a branch point p when Σ is topologically embedded. It defines a link L(p), the components of which are closed braids with the same axis up to orientation. If Σ is closed without boundary, the contribution of p to the degree of the normal bundle of Σ in M can be computed on the link L(p), in terms of the algebraic crossing numbers of its components and of their linking numbers with one another.      

Counting curves via lattice paths in polygons

This Note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface is $\text{P}{}^2$ or $\text{P}{}^1\text{×}\text{P}{}^1$ then the invariants under consideration coincide with the Gromov–Witten invariants. The formula gives a new count even in these cases, where other computational techniques are available. To cite this article: G. Mikhalkin, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On cubic surfaces with a rational line

We report on our project to construct non-singular cubic surfaces over \mathbbQ{\mathbb{Q}} with a rational line. Our method is to start with degree 4 Del Pezzo surfaces in diagonal form. For these, we develop an explicit version of Galois descent.