Advanced Studies in Pure Mathematics Moduli Spaces and Arithmetic Geometry Kyoto pp

Arnaud Beauville

Description

Niveau: Supérieur, Doctorat, Bac+8
Advanced Studies in Pure Mathematics 45, 2006 Moduli Spaces and Arithmetic Geometry (Kyoto 2004) pp. 145–156 Vector bundles on curves and theta functions Arnaud Beauville Abstract. This is a survey lecture on the “theta map” from the moduli space of SLr-bundles on a curve C to the projective space of r-th order theta functions on JC. Some recent results and a few open problems about that map are discussed. Introduction These notes survey the relation between the moduli spaces of vector bundles on a curve C and the spaces of (classical) theta functions on the Jacobian J of C. The connection appears when one tries to describe the moduli space Mr of rank r vector bundles with trivial determinant as a projective variety in an explicit way (as opposed to the somewhat non-constructive way provided by GIT). The Picard group of the moduli space is infinite cyclic, generated by the determinant line bundle L ; thus the natural maps from Mr to projective spaces are those defined by the linear systems |Lk|, and in the first instance the map ?L : Mr |L|?. The key point is that this map can be identified with the theta map ? : Mr |r?| which associates to a general bundle E ? Mr its theta divisor ?E , an element of the linear system |r?| on J – we will recall the precise defi- nitions below.

• has usually


• large base


• rank


• no theta


• locus consists


• theta map


• base locus


• raynaud bundle


• slope µ

Sujets

• Beauville
• Base locus

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 11
• Langue: English

Extrait

AdvancedStudiesinPureMathematics45,2006 ModuliSpacesandArithmeticGeometry(Kyoto2004) pp.145–156 Vectorbundlesoncurvesandthetafunctions ArnaudBeauville Abstract. Thisisasurveylectureonthe“thetamap”fromthemodulispaceof SL r -bundlesonacurveCtotheprojectivespaceofr-thorderthetafunctions onJC.Somerecentresultsandafewopenproblemsaboutthatmapare discussed. Introduction Thesenotessurveytherelationbetweenthemodulispacesofvector bundlesonacurveCandthespacesof(classical)thetafunctionsonthe JacobianJofC.Theconnectionappearswhenonetriestodescribe themodulispace M r ofrankrvectorbundleswithtrivialdeterminant asaprojectivevarietyinanexplicitway(asopposedtothesomewhat non-constructivewayprovidedbyGIT).ThePicardgroupofthemoduli spaceisinfinitecyclic,generatedbythedeterminantlinebundle L ;thus thenaturalmapsfrom M r toprojectivespacesarethosedefinedbythe linearsystems |L k | ,andinthefirstinstancethemapϕ L : M r

|L| ∗ . Thekeypointisthatthismapcanbeidentifiedwiththethetamap θ: M r

| rΘ | whichassociatestoageneralbundleE ∈M r itsthetadivisorΘ E ,an elementofthelinearsystem | rΘ | onJ–wewillrecalltheprecisedefi- nitionsbelow.Thisdescriptionturnsouttobesufficientlymanageable togetsomeinformationonthebehaviourofthismap,atleastwhenr orgaresmall. Wewilldescribetheresultswhichhavebeenobtainedsofar–most ofthemfairlyrecently.Thusthesenotescanbeviewedasasequelto [B2],thoughwithamoreprecisefocusonthethetamap.Forthecon- venienceofthereaderwehavemadethispaperindependentof[B2],by recallingin § 1thenecessarydefinitions.Thenwediscusstheindetermi- nacylocusofθ( § 2),thecaser=2( § 3),thecaseg=2( § 4),andthe ReceivedMarch9,2005 RevisedMarch29,2006