Quadratic Differentials and Equivariant Deformation Theory of Curves

Bernhard Köck; Aristides Kontogeorgis

Displaying similar documents to “Quadratic Differentials and Equivariant Deformation Theory of Curves”

An Algebraic Formula for the Index of a Vector Field on an Isolated Complete Intersection Singularity

H.-Ch. Graf von Bothmer, Wolfgang Ebeling, Xavier Gómez-Mont (2008)

Annales de l’institut Fourier

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Let $(V, 0)$ be a germ of a complete intersection variety in $ℂ^{n + k}$ , $n > 0$ , having an isolated singularity at $0$ and $X$ be the germ of a holomorphic vector field having an isolated zero at $0$ and tangent to $V$ . We show that in this case the homological index and the GSV-index coincide. In the case when the zero of $X$ is also isolated in the ambient space $ℂ^{n + k}$ we give a formula for the homological index in terms of local linear algebra.

The higher transvectants are redundant

Abdelmalek Abdesselam, Jaydeep Chipalkatti (2009)

Annales de l’institut Fourier

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Let $A, B$ denote generic binary forms, and let $𝔲_{r} = {(A, B)}_{r}$ denote their $r$ -th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the ${𝔲_{r}}$ . As a consequence, we show that each of the higher transvectants ${𝔲_{r} : r \geq 2}$ is redundant in the sense that it can be completely recovered from $𝔲_{0}$ and $𝔲_{1}$ . This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of $S L_{2}$ -representations,...

Computing limit linear series with infinitesimal methods

Laurent Evain (2007)

Annales de l’institut Fourier

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Alexander and Hirschowitz determined the Hilbert function of a generic union of fat points in a projective space when the number of fat points is much bigger than the greatest multiplicity of the fat points. Their method is based on a lemma which determines the limit of a linear system depending on fat points approaching a divisor. Other Hilbert functions were computed previously by Nagata. In connection with his counter-example to Hilbert’s fourteenth problem, Nagata determined...

On Halphen’s Theorem and some generalizations

Alcides Lins Neto (2006)

Annales de l’institut Fourier

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Let $M^{n}$ be a germ at $0 \in ℂ^{m}$ of an irreducible analytic set of dimension $n$ , where $n \geq 2$ and $0$ is a singular point of $M$ . We study the question: when does there exist a germ of holomorphic map $φ : (ℂ^{n}, 0) \to (M, 0)$ such that $φ^{- 1} (0) = {0}$ ? We prove essentialy three results. In Theorem 1 we consider the case where $M$ is a quasi-homogeneous complete intersection of $k$ polynomials $F = (F_{1}, ..., F_{k})$ , that is there exists a linear holomorphic vector field $X$ on $ℂ^{m}$ , with eigenvalues $λ_{1}, ..., λ_{m} \in ℚ_{+}$ such that $X (F^{T}) = U \cdot F^{T}$ , where $U$ is a $k \times k$ matrix with entries in $𝒪_{m}$ . We prove that if...

An explicit formula for the Hilbert symbol of a formal group

Floric Tavares Ribeiro (2011)

Annales de l’institut Fourier

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A Brückner-Vostokov formula for the Hilbert symbol of a formal group was established by Abrashkin under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ( $ϕ, Γ$ )-modules and a cohomological interpretation of Abrashkin’s technique. To do this, we build ( $ϕ, Γ$ )-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas...