Lundi 19 octobre 2009
Freiburg-Nancy-Strasbourg Joint Seminar in Algebraic and Complex Geometry.
Lundi 19 octobre 2009
10:00 - 11:00
Sebastian Neumann, Freiburg
On the Harder-Narasimhan filtration of the tangent bundleGiven an ample line bundle on a projective manifoldthere exists a filtration of the tangent bundlethe Harder-Narasimhan filtrationwhich heavily depends on the choice of the ample bundleIf the subsheaves of the filtration satisfy some positivity conditionthey give rise to rationally connected foliationsWe explainhow this filtration depends on the chosen ample line bundlehow the subsheaves in the filtration can be characterized on Fano 3-folds and discuss the connection to the maximal rationally connected quotient.
11:15 - 12:15
Sönke Rollenske, Bonn
Compact moduli for certain Kodaira fibrationsThe modular compactification of the moduli space of smooth curves (of genus at least 2via stable curves is by now classicalFor higher dimensional analogs of this constructionfirst proposed by Kollár and Shepherd-Barronmuch less is known and several technical issues ariseWe will give a short introduction and then show how to describe explicitly the surfaces occuring as stable degenerations of some surfaces of general type.
14:00 - 15:00
Ulrich Derenthal, Freiburg
Counting rational points via universal torsorsFano varietiesfor example cubic surfacesoften contain infinitely many rational pointsA conjecture of Manin predicts their distribution preciselyOne approach to this conjecture uses a parametrization of the rational points on the Fano variety by integral points on certain higher-dimensional varieties called universal torsorsFor certain singular cubic surfacesthis leads to a proof of Manin’s conjecture.
16:00 - 17:00
Gianluca Pacienza, IRMA
On volumes along subvarieties of line bundles with non-negative Kodaira-Iitaka dimensionThe restricted volume has played an increasingly important role in the asymptotic study of linear seriesThis invariant measures how many sections of (the multiples ofa line bundle restricted to a subvariety can be lifted to the ambient varietyIts behavior is well-understood in the big caseIn a joint work with STakayama we study the restricted volume along subvarieties of line bundles with non-negative Kodaira-Iitaka dimensionOur main interest is to compare it with a similar notion defined in terms of the asymptotic multiplier ideal sheafwith which it coincides in the big caseWe shall prove that the former is non-zero if and only if the latter isWe then study inequalities between them and prove that if they coincide on every very general curve the line bundle must have zero Kodaira-Iitaka dimension or be big.