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A limit linear series moduli scheme

Brian Osserman (2006)

Annales de l’institut Fourier

We develop a new, more functorial construction for the basic theory of limit linear series, which provides a compactification of the Eisenbud-Harris theory. In an appendix, in order to obtain the necessary dimensional lower bounds on our limit linear series scheme we develop a theory of “linked Grassmannians”; these are schemes parametrizing sub-bundles of a sequence of vector bundles, which map into one another under fixed maps of the ambient bundles.

A metric graph satisfying [...] w 4 1 = 1 w 4 1 = 1 that cannot be lifted to a curve satisfying [...] dim ⁡   ( W 4 1 ) = 1 dim ( W 4 1 ) = 1

Marc Coppens (2016)

Open Mathematics

For all integers g ≥ 6 we prove the existence of a metric graph G with [...] w41=1 w 4 1 = 1 such that G has Clifford index 2 and there is no tropical modification G′ of G such that there exists a finite harmonic morphism of degree 2 from G′ to a metric graph of genus 1. Those examples show that not all dimension theorems on the space classifying special linear systems for curves have immediate translation to the theory of divisors on metric graphs.

ACM embeddings of curves of a quadric surface

S. Giuffrida, R. Maggioni, R. Re (2007)

Collectanea Mathematica

Let C be a smooth integral projective curve admitting two pencils ga1 and gb1 such that ga1 + gb1 is birational. We give conditions in order that the complete linear system |sga1 + rgb1| be normally generated or very ample.

Brill–Noether loci for divisors on irregular varieties

Margarida Mendes Lopes, Rita Pardini, Pietro Pirola (2014)

Journal of the European Mathematical Society

We take up the study of the Brill-Noether loci W r ( L , X ) : = { η Pic 0 ( X ) | h 0 ( L η ) r + 1 } , where X is a smooth projective variety of dimension > 1 , L Pic ( X ) , and r 0 is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for h 0 ( K D ) , where D is a divisor that moves linearly on a smooth projective variety X of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension > 2 . In the 2 -dimensional case we prove an...

Clifford’s Theorem for real algebraic curves

Jean-Philippe Monnier (2010)

Annales de l’institut Fourier

We establish, for smooth projective real curves, an analogue of the classical Clifford inequality known for complex curves. We also study the cases when equality holds.

Codimension 1 subvarieties g and real gonality of real curves

Edoardo Ballico (2003)

Czechoslovak Mathematical Journal

Let g be the moduli space of smooth complex projective curves of genus g . Here we prove that the subset of g formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in g . As an application we show that if X g is defined over , then there exists a low degree pencil u X 1 defined over .

Curves with only triple ramification

Stefan Schröer (2003)

Annales de l'Institut Fourier

We show that the set of smooth curves of genus g 0 admitting a branched covering X 1 with only triple ramification points is of dimension at least max ( 2 g - 3 , g ) . In characteristic two, such curves have tame rational functions and an analog of Belyi’s Theorem applies to them.

Energy decay rates for solutions of Maxwell's system with a memory boundary condition

Serge Nicaise, Cristina Pignotti (2007)

Collectanea Mathematica

We consider the stabilization of Maxwell's equations with space variable coefficients in a bounded region with a smooth boundary, subject to dissipative boundary conditions of memory type on the boundary. Under suitable conditions on the domain and on the permeability and permittivity coefficients, we prove the exponential/polynomial decay of the energy. Our result is mainly based on the use of the multipliers method and the introduction of a suitable Lyapounov functional.

Five-gonal curves of genus nine.

Marc Coppens (2005)

Collectanea Mathematica

Let C be a smooth 5-gonal curve of genus 9. Assume all linear systems g15 on C are of type I (i.e. they can be counted with multiplicity 1) and let m be the numer of linear systems g15 on C. The only possibilities are m=1; m=2; m=3 and m=6. Each of those possibilities occur.

Gonality for stable curves and their maps with a smooth curve as their target

Edoardo Ballico (2009)

Open Mathematics

Here we study the deformation theory of some maps f: X → ℙr , r = 1, 2, where X is a nodal curve and f|T is not constant for every irreducible component T of X. For r = 1 we show that the “stratification by gonality” for any subset of [...] with fixed topological type behaves like the stratification by gonality of M g.

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 .

La conjecture de Green générique

Arnaud Beauville (2003/2004)

Séminaire Bourbaki

Une courbe C projective et lisse de genre g , non hyperelliptique, admet un plongement canonique dans un espace projectif g - 1 . Un résultat classique affirme que l’idéal gradué I C des équations de C dans g - 1 est engendré par ses éléments de degré 2 , sauf si C admet certains systèmes linéaires très particuliers. Mark Green en a proposé il y a vingt ans une vaste généralisation, qui décrit la résolution minimale de I C en fonction de l’existence de systèmes linéaires spéciaux sur C . Claire Voisin vient de...

Lie description of higher obstructions to deforming submanifolds

Marco Manetti (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

To every morphism χ : L M of differential graded Lie algebras we associate a functors of artin rings Def χ whose tangent and obstruction spaces are respectively the first and second cohomology group of the suspension of the mapping cone of χ . Such construction applies to Hilbert and Brill-Noether functors and allow to prove with ease that every higher obstruction to deforming a smooth submanifold of a Kähler manifold is annihilated by the semiregularity map.

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