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Even sets of nodes on sextic surfaces

Fabrizio Catanese, Fabio Tonoli (2007)

Journal of the European Mathematical Society

We determine the possible even sets of nodes on sextic surfaces in ℙ 3 , showing in particular that their cardinalities are exactly the numbers in the set { 24 , 32 , 40 , 56 } . We also show that all the possible cases admit an explicit description. The methods that we use are an interplay of coding theory and projective geometry on one hand, and of homological and computer algebra on the other. We give a detailed geometric construction for the new case of an even set of 56 nodes, but the ultimate verification of existence...

Felix Klein's paper on real flexes vindicated

Felice Ronga (1998)

Banach Center Publications

In a paper written in 1876 [4], Felix Klein gave a formula relating the number of real flexes of a generic real plane projective curve to the number of real bitangents at non-real points and the degree, which shows in particular that the number of real flexes cannot exceed one third of the total number of flexes. We show that Klein's arguments can be made rigorous using a little of the theory of singularities of maps, justifying in particular his resort to explicit examples.

Invariants of real symplectic four-manifolds out of reducible and cuspidal curves

Jean-Yves Welschinger (2006)

Bulletin de la Société Mathématique de France

We construct invariants under deformation of real symplectic four-manifolds. These invariants are obtained by counting three different kinds of real rational J -holomorphic curves which realize a given homology class and pass through a given real configuration of (the appropriate number of) points. These curves are cuspidal curves, reducible curves and curves with a prescribed tangent line at some real point of the configuration. They are counted with respect to some sign defined by the parity of...

Invertible cohomological field theories and Weil-Petersson volumes

Yuri I. Manin, Peter Zograf (2000)

Annales de l'institut Fourier

We show that the generating function for the higher Weil–Petersson volumes of the moduli spaces of stable curves with marked points can be obtained from Witten’s free energy by a change of variables given by Schur polynomials. Since this generating function has a natural extension to the moduli space of invertible Cohomological Field Theories, this suggests the existence of a “very large phase space”, correlation functions on which include Hodge integrals studied by C. Faber and R. Pandharipande....

Labeled floor diagrams for plane curves

Sergey Fomin, Grigory Mikhalkin (2010)

Journal of the European Mathematical Society

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) Gromov–Witten 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...

Monotone Hurwitz Numbers and the HCIZ Integral

I. P. Goulden, Mathieu Guay-Paquet, Jonathan Novak (2014)

Annales mathématiques Blaise Pascal

In this article, we prove that the complex convergence of the HCIZ free energy is equivalent to the non-vanishing of the HCIZ integral in a neighbourhood of z = 0 . Our approach is based on a combinatorial model for the Maclaurin coefficients of the HCIZ integral together with classical complex-analytic techniques.

Currently displaying 61 – 80 of 175