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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.

Finiteness of cominuscule quantum K -theory

Anders S. Buch, Pierre-Emmanuel Chaput, Leonardo C. Mihalcea, Nicolas Perrin (2013)

Annales scientifiques de l'École Normale Supérieure

The product of two Schubert classes in the quantum K -theory ring of a homogeneous space X = G / P is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on  X . We show that if X is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to  X that take the marked points to general Schubert varieties and whose domains...

Freeness of hyperplane arrangements and related topics

Masahiko Yoshinaga (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

These are the expanded notes of the lecture by the author in “Arrangements in Pyrénées”, June 2012. We are discussing relations of freeness and splitting problems of vector bundles, several techniques proving freeness of hyperplane arrangements, K. Saito’s theory of primitive derivations for Coxeter arrangements, their application to combinatorial problems and related conjectures.

Generalized polar varieties and an efficient real elimination

Bernd Bank, Marc Giusti, Joos Heintz, Luis M. Pardo (2004)

Kybernetika

Let W be a closed algebraic subvariety of the n -dimensional projective space over the complex or real numbers and suppose that W is non-empty and equidimensional. In this paper we generalize the classic notion of polar variety of W associated with a given linear subvariety of the ambient space of W . As particular instances of this new notion of generalized polar variety we reobtain the classic ones and two new types of polar varieties, called dual and (in case that W is affine) conic. We show that...

Geometric structures on the complement of a projective arrangement

Wim Couwenberg, Gert Heckman, Eduard Looijenga (2005)

Publications Mathématiques de l'IHÉS

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= finite union of hyperplanes) whose Levi-Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements defined by finite complex reflection groups. We determine a parameter interval for which the metric is locally of Fubini-Study type, flat, or complex-hyperbolic. We find a finite subset of this interval for...

Grassmann defective surfaces

Claudio Fontanari (2004)

Bollettino dell'Unione Matematica Italiana

A projective variety V is 1 , h -defective if the Grassmannian of lines contained in the span of h + 1 independent points on V has dimension less than the expected one. In the present paper, which is inspired by classical work of Alessandro Terracini, we prove a criterion of 1 , h -defectivity for algebraic surfaces and we discuss its applications to Veronese embeddings and to rational normal scrolls.

Green's generic syzygy conjecture for curves of even genus lying on a K3 surface

Claire Voisin (2002)

Journal of the European Mathematical Society

We consider the generic Green conjecture on syzygies of a canonical curve, and particularly the following reformulation thereof: For a smooth projective curve C of genus g in characteristic 0, the condition Cliff C > l is equivalent to the fact that K g - l ' - 2 , 1 ( C , K C ) = 0 , ∀ l ' ≤ l . We propose a new approach, which allows up to prove this result for generic curves C of genus g ( C ) and gonality gon(C) in the range g ( C ) 3 + 1 ≤ gon(C) ≤ g ( C ) 2 + 1 .

Gromov–Witten invariants for mirror orbifolds of simple elliptic singularities

Ikuo Satake, Atsushi Takahashi (2011)

Annales de l’institut Fourier

We consider a mirror symmetry of simple elliptic singularities. In particular, we construct isomorphisms of Frobenius manifolds among the one from the Gromov–Witten theory of a weighted projective line, the one from the theory of primitive forms for a universal unfolding of a simple elliptic singularity and the one from the invariant theory for an elliptic Weyl group. As a consequence, we give a geometric interpretation of the Fourier coefficients of an eta product considered by K. Saito.

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