Absolutely split real algebraic vector bundles over a real form of projective space

Let M be a projective variety, defined over the field of real numbers, with the property that the base change MR_×C is isomorphic to CPN for some N. A real algebraic vector bundle E over M will be called absolutely split if the vector bundle ER_⊗C over MR_×C splits into a direct sum of line bundles. We classify the isomorphism classes of absolutely split vector bundles over M.     

Modules over twisted group rings and vector bundles over the anisotropic real conic

We prove, in an elementary way, that a locally free sheaf of finite rank over the anisotropic real conic is the direct sum of indecomposable locally free sheaves of rank 1 or 2. Our proof is purely algebraic, and is based on a classification of graded ?[X, Y]-modules endowed with a certain action of the cyclic group ?/4?.     

Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n.     

Fixed points of automorphisms of real algebraic curves

We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of an automorphism and the maximum order of an abelian group of automorphisms of a real curve. We also bound the full group of automorphisms of a real hyperelliptic curve. Work supported by the European Community’s Human Potential Programme under contract HPRN-CT-2001-00271, RAAG.     

Numerically effective line bundles associated to a stable bundle over a curve

Let G be a complex semisimple group and χ a character of a parabolic subgroup P⊂G such that the associated line bundle on G/P is ample. For a general stable G-bundle E_G over a compact Riemann surface of genus at least two, the line bundle over E_G/P defined by χ has the property that the restriction of  to any closed subvariety of E_G/P of smaller dimension is ample, although is not ample.     

Imaginary automorphisms on real hyperelliptic curves

A real hyperelliptic curve X is said to be Gaussian if there is an automorphism such that , where [-1] denotes the hyperelliptic involution on X. Gaussian curves arise naturally in several contexts, for example when one studies real Jacobians. In the present paper, we study the properties of Gaussian curves and we describe their moduli spaces.     

Second order homological obstructions on real algebraic manifolds

Let Y be a compact nonsingular real algebraic set whose homology classes (over Z/2) are represented by Zariski closed subsets. It is well known that every smooth map from a compact smooth manifold to Y is unoriented bordant to a regular map. In this paper, we show how to construct smooth maps from compact nonsingular real algebraic sets to Y not homotopic to any regular map starting from a nonzero homology class of Y of positive degree. We use these maps to obtain obstructions to the existence of local algebraic tubular neighborhoods of algebraic submanifolds of Rn and to study some algebro-homological properties of rational real algebraic manifolds.     

Jets of line bundles on curves and Wronskians

We collect a few results about jets of line bundles on curves and Wronskians, with a special emphasis to those arising from the canonical involution of a hyperelliptic curve.     

Witt equivalence of algebraic function fields over real closed fields

We examine the conditions for two algebraic function fields over real closed fields to be Witt equivalent. We show that there are only two Witt classes of algebraic function fields with a fixed real closed field of constants: real and non-real ones. The first of them splits further into subclasses corresponding to the tame equivalence. This condition has a natural interpretation in terms of both: orderings (the associated Harrison isomorphism maps 1-pt fans onto 1-pt fans), and geometry and topology of associated real curves (the bijection of points is a homeomorphism and these two curves have the same number of semi-algebraically connected components). Finally, we derive some immediate consequences of those theorems. In particular we describe all the Witt classes of algebraic function fields of genus 0 and 1 over the fixed real closed field. Received: 16 February 2000; in final form: 7 December 2000 / Published online: 18 January 2002     

Normally generated line bundles of smallest degree on curves

In this paper we study the embeddings of smallest degree of a generic curveC of genusg as a projectively normal curve.

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Sunto In questo articolo studiamo le immersioni di grado minimo di una generica curvaC di genereg come curva proiettivamente normale.

     

Computing complex and real tropical curves using monodromy

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are computed using an implementation that builds on the numerical algebraic geometry software Bertini.     

Double coverings of hyperelliptic real algebraic curves

We consider double and (possibly) branched coverings π:X→X^′ between real algebraic curves where X is hyperelliptic. We are interested in the topology of such coverings and also in describing them in terms of algebraic equations. In this article we completely solve these two problems. We first analyse the topological features and ramification data of such coverings. Second, for each isomorphism class of these coverings we then describe a representative, with defining polynomial equations for X and for X^′, a formula for the involution that generates the covering transformation group, and a rational formula for the covering projection π:X→X^′.     

Rokhlin's inequalities for real algebraic curves and¶an extremal property

Rokhlin's inequalities for real algebraic curves and an extremal property for one of the inequalities are derived via a link cobordism approach. Received: 18 February 1997 / Revised version: 23 September 1998     

On globally generated vector bundles on projective spaces II

Extending the main result of Sierra and Ugaglia (2009)  [12], we classify globally generated vector bundles on Pⁿ${\mathbb{P}}^n$ with first Chern class equal to 3.     

Remarks on normal generation of line bundles on algebraic curves

We determine necessary and sufficient conditions for nonspecial line bundles of degree 2% - 4 and 2g - 5 being not normally generated. Furthermore, we also determine necessary and suffcient conditions for speciality 1 line bundles of degree 2g -7,2% - 8, and 2g - 9 being not normally generated.