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LABELED FLOOR DIAGRAMS FOR PLANE CURVES
, 906
"... Abstract. Floor diagrams are a class of weighted oriented graphs introduced by E. Brugallé and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) GromovWitten invariants of projective spaces in terms of floor diagrams and their generalizatio ..."
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Abstract. Floor diagrams are a class of weighted oriented graphs introduced by E. Brugallé and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) GromovWitten invariants of projective spaces in terms of floor diagrams and their generalizations. In a number of cases, these descriptions can be used to obtain explicit (direct or recursive) formulas for the corresponding enumerative invariants. In particular, we use this approach to enumerate rational curves of given degree passing through a collection of points on the complex plane and having maximal tangency to a given line. Another application of the combinatorial approach is a proof of a conjecture by P. Di Francesco–C. Itzykson and L. Göttsche that in the case of a fixed cogenus, the number of plane curves of degree d passing through suitably many generic points is given by a polynomial in d, assuming that d is sufficiently large. Furthermore, the proof provides a method for computing these “node polynomials.” A labeled floor diagram is obtained by labeling the vertices of a floor diagram by the integers 1,...,d in a manner compatible with the orientation. We show that
Universal Polynomials for Severi Degrees of Toric Surfaces
"... Abstract. The Severi variety parameterizes plane curves of degree d with δ nodes. Its degree is called the Severi degree. For large enough d, the Severi degrees coincide with the GromovWitten invariants of CP 2. Fomin and Mikhalkin (2009) proved the 1995 conjecture that for fixed δ, Severi degrees ..."
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Abstract. The Severi variety parameterizes plane curves of degree d with δ nodes. Its degree is called the Severi degree. For large enough d, the Severi degrees coincide with the GromovWitten invariants of CP 2. Fomin and Mikhalkin (2009) proved the 1995 conjecture that for fixed δ, Severi degrees are eventually polynomial in d. In this paper, we study the Severi varieties corresponding to a large family of toric surfaces. We prove the analogous result that the Severi degrees are eventually polynomial as a function of the multidegree. More surprisingly, we show that the Severi degrees are also eventually polynomial “as a function of the surface”. Our strategy is to use tropical geometry to express Severi degrees in terms of Brugallé and Mikhalkin’s floor diagrams, and study those combinatorial objects in detail. An important ingredient in the proof is the polynomiality of the discrete volume of a variable facetunimodular polytope. Résumé. La variété de Severi paramétrise les courbes planes de degré d avec δ noeuds. Son degré s’appelle le degré de Severi. Pour d assez grand, les degrés de Severi coïncident avec les invariants de GromovWitten de CP 2. Fomin et Mikhalkin (2009) ont prouvé une conjecture de 1995 que pour δ fixé, les degrés de Severi sont à terme des polynômes en d. Nous étudions les variétés de Severi correspondant à une large famille de surfaces toriques. Nous prouvons le résultat analogue que les degrés de Severi sont à terme des fonctions polynomiales du multidegré. De manière plus surprenante,
On the geometry of some strata of unisingular curves
, 2008
"... We study geometric properties of linear strata of unisingular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection ..."
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We study geometric properties of linear strata of unisingular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection rings of the closures of the strata. The rational equivalence classes of some geometric cycles on the strata are calculated. As an application we give an example when the proper stratum is not affine. As an auxiliary problem we discuss the collision of two singular points, restrictions on possible resulting singularity types and solve the collision problem in several cases. Then we present some cases of enumeration of
1 On the enumeration of complex plane curves with two singular points
, 2008
"... We study equisingular strata of curves with two singular points of prescribed types. The method of the previous work [Kerner04] is generalized to this case. This allows to solve the enumerative problem for plane curves with two singular points of linear singularity types. In the general case this r ..."
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We study equisingular strata of curves with two singular points of prescribed types. The method of the previous work [Kerner04] is generalized to this case. This allows to solve the enumerative problem for plane curves with two singular points of linear singularity types. In the general case this reduces the enumerative questions to the problem of collision of the two singular points. The method is applied to several cases, e.g. enumeration of curves with two ordinary multiple points, with a point of a linear singularity type and a node etc. Explicit numerical results are given. An elementary application of the method is the determination of Thom polynomials for curves with one singular point (for some series of singularity types). Some examples are given. MSC: primary14N10, 14N35 secondary14H10, 14H50
Enriques diagrams, infinitely near points, and Hilbert schemes
, 905
"... with an appendix by Ilya TYOMKIN Abstract. We study sequences of infinitely near Tpoints of a smooth family F/Y of geometrically irreducible surfaces. We destinguish a special sort of such sequences, the strict sequences. To each one, we associate an ordered unweighted Enriques diagram. We prove th ..."
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with an appendix by Ilya TYOMKIN Abstract. We study sequences of infinitely near Tpoints of a smooth family F/Y of geometrically irreducible surfaces. We destinguish a special sort of such sequences, the strict sequences. To each one, we associate an ordered unweighted Enriques diagram. We prove that the various sequences with a given diagram form a functor, and we represent it by a smooth Yscheme. We equip this scheme with a free action of the automorphism group of the diagram. We equip the diagram with weights, take the subgroup of automorphisms preserving the weights, and form the corresponding quotient scheme. Our main theorem asserts the existence of a universally injective map from this quotient scheme to the Hilbert scheme of F/Y; further, this map is an embedding in characteristic 0. However, in every positive characteristic, we give an example where the map is purely inseparable. 1.
Node Polynomials
"... Abstract. According to the Göttsche conjecture (now a theorem), the degree N d,δ of the Severi variety of plane curves of degree d with δ nodes is given by a polynomial in d, provided d is large enough. These “node polynomials” Nδ(d) were determined by Vainsencher and Kleiman–Piene for δ ≤ 6 and δ ≤ ..."
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Abstract. According to the Göttsche conjecture (now a theorem), the degree N d,δ of the Severi variety of plane curves of degree d with δ nodes is given by a polynomial in d, provided d is large enough. These “node polynomials” Nδ(d) were determined by Vainsencher and Kleiman–Piene for δ ≤ 6 and δ ≤ 8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute Nδ(d) for δ ≤ 14. Furthermore, we improve the threshold of polynomiality and verify Göttsche’s conjecture on the optimal threshold up to δ ≤ 14. We also determine the first 9 coefficients of Nδ(d), for general δ, settling and extending a 1994 conjecture of Di Francesco and Itzykson. Résumé. Selon la Conjecture de Göttsche (maintenant un Théorème), le degré N d,δ de la variété de Severi des courbes planes de degré d avec δ noeuds est donné par un polynôme en d, pour d assez grand. Ces polynômes de noeuds Nδ(d) ont été déterminés par Vainsencher et Kleiman–Piene pour δ ≤ 6 et δ ≤ 8, respectivement. S’appuyant sur les idées de Fomin et Mikhalkin, nous développons un algorithme explicite permettant de calculer tous les polynômes de noeuds, et l’utilisons pour calculer Nδ(d), pour δ ≤ 14. De plus, nous améliorons le seuil de polynomialité et vérifions la Conjecture de Göttsche sur le seuil optimal jusqu’à δ ≤ 14. Nous déterminons aussi les 9 premiers coéfficients de Nδ(d), pour un δ quelconque, confirmant et étendant la Conjecture de Di Francesco et Itzykson de 1994.
Relative Node Polynomials for Plane Curves
"... Abstract. We generalize the recent work of Fomin and Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generalized Severi var ..."
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Abstract. We generalize the recent work of Fomin and Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial in d, provided δ is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined “relative node polynomial ” in the tangency orders, provided the latter are large enough. We describe a method to compute these polynomials for arbitrary δ, and use it to present explicit formulas for δ ≤ 6. We also give a threshold for polynomiality, and compute the first few leading terms for any δ. Résumé. Nous généralisons les travaux récents de Fomin et Mikhalkin sur des formules polynomiales pour les degrés de Severi. Le degré de la variété de Severi des courbes planes de degré d et à δ nœuds est donné par un polynôme en d, pour δ fixé et d assez grand. Nous étendons ce résultat aux variétés de Severi généralisées paramétrant les courbes planes et qui, en outre, satisfont à des conditions de tangence d’ordres donnés avec une droite fixée. Nous montrons que les degrés de ces variétés, rééchelonnés de manière appropriée, sont donnés par un “polynôme de noeud relatif”, défini combinatoirement, en les ordres de tangence, dès que ceuxci sont assez grands. Nous décrivons une méthode pour calculer ces polynômes pour delta arbitraire, et l’utilisons pour présenter des formules explicites pour δ ≤ 6. Nous donnons aussi un seuil pour la polynomialité, et calculons les premiers termes dominants pour tout δ.