Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight 2. Moreover, we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and we study Faltings heights of arithmetic Hirzebruch-Zagier divisors.

El teorema de Riemann-Roch y el morfismo de Gysin en geometría aritmética

Le th eor eme de Riemann-Roch originale a rme que pour tout morphisme propre f : Y ! X entre vari et es quasi-projectifs lisses sur un corps, et tout el ement a 2 K0(Y ) du groupe de Grothendieck des br es vectoriels on a ch(f!(a)) = f {u100000}Td(Tf ) ch(a) (cf. [BS58]). Ici ch est le caract ere de Chern, Td(Tf ) est la classe de Todd du br e tangent relative et f et f! sont les images directes de l'anneau de Chow et K0 respectivement. Apr es, Baum, Fulton et MacPherson ont d emontr e en [BFM75] le th eor eme de Riemann-Roch pour des morphismes localement intersection compl ete entre des sch emas alg ebriques (sch emas s epar es et localement de type ni sur un corps) projectifs et singuli eres. En [FG83] Fulton et Gillet ont d emontr e le th eor eme sans hypoth eses projectifs. L'extension a la th eorie K sup erieure pour des sch emas r eguli eres sur une base fut d emontr e par Gillet en [Gil81]. Le th eor eme de Riemann-Roch qu'il prouve est pour des morphismes projectifs entre des sch emas lisses et quasi-projectifs. Donc, dans le cas des sch emas sur un corps, le r esultat de Gillet n'inclus pas le th eor eme de [BFM75]. La plus grande g en eralisation du th eor eme de Riemann-Roch que je connais est [D eg14] et [HS15], o u D eglise et Holmstrom-Scholbach obtiennent ind ependamment le th eor eme de Riemann- Roch pour la K-th eorie sup erieure et les morphismes projectifs lic entre sch emas r eguli eres sur une base noetherienne de dimension nie... NOTA 520 8 El teorema de Riemann-Roch original de Grothendieck a rma que para todo mor smo propio f : Y ! X, entre variedades irreducibles quasiproyectivas lisas sobre un cuerpo, y todo elemento a 2 K0(Y ) del grupo de Grothendieck de brados vectoriales se satisface la relaci on ch(f!(a)) = f {u100000}Td(Tf ) ch(a) (cf. [BS58]). Recu erdese que ch denota el car acter de Chern, Td(Tf ) la clase de Todd del brado tangente relativo y f y f! las im agenes directas en el anillo de Chow y K0 respectivamente. M as tarde Baum, Fulton MacPherson probaron en [BFM75] el teorema de Riemann-Roch para mor smos localmente intersecci on completa entre esquemas algebraicos (es decir, esquemas separados localmente de tipo nito sobre cuerpo) proyectivos singulares. En [FG83] Fulton y Gillet probaron el teorema sin hip otesis proyectivas. La notable extensi on a la teor a K superior para esquemas regulares sobre una base fue probada por Gillet en [Gil81]. El teorema de Riemann-Roch all probado es para mor smos proyectivos entre esquemas lisos quasiproyectivos. Sin embargo, obs ervese que en el caso de esquemas sobre cuerpo el resultado de Gillet no recupera el teorema de [BFM75]. La mayor generalizaci on del teorema de Riemann-Roch que yo conozco es [D eg14] y [HS15] donde D eglise y Holmstrom-Scholbach obtuvieron independientemente el teorema de Riemann-Roch para teor a K superior y mor smos proyectivos lic entre esquemas regulares sobre una base noetheriana nito dimensional...

Geometria e arirmética numa visão multicultural: uma experiência pedagógica

This paper aims to build a notebook of activities that can help the teacher of elementary school mathematics. Topics covered are arithmetic and geometry and the activities proposed here were developed aiming print them a multicultural character. We take as a base line developed by Claudia Zaslavsky multiculturalism and reflected in his books "Games and activities worldwide" and "More games and activities worldwide." We structure our work around four themes: the symbol of the Olympic Games, the pyramids of Egypt, the Russian abacus abacus and Chinese. The first two themes allow you to explore basic concepts of geometry while the latter two themes allow us to explore numerical notation and arithmetic operations

Contractions in the 2-wasserstein lenght space and thermalization of granular media

An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite-dimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even non-convexity) | if uniformly controlled | will quantify contractivity (limit expansivity) of the flow.

Elements of algebraic geometry and the positive theory of partially commutative groups

The first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theories of the direct factors of G (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group H. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of H has quantifier elimination and that arbitrary first-order formulas lift from H to H * F, where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.

Aspectes geomètrics i dinàmics de transformacions racionals planes

Donada una aplicació racional en una varietat complexa, Bellon i Viallet van definit l’entropia algebraica d’aquesta aplicació i van provar que aquest valor és un invariant biracional. Un invariant biracional equivalent és el grau asimptòtic, grau dinàmic o complexitat, definit per Boukraa i Maillard. Aquesta noció és propera a la complexitat definida per Arnold. Conjecturalment, el grau asimptòtic satisfà una recurrència lineal amb coeficients enters. Aquesta conjectura ha estat provada en el cas polinòmic en el pla afí complex per Favre i Jonsson i resta oberta en per al cas projectiu global i per al cas local. L’estudi de l’arbre valoratiu de Favre i Jonsson ha resultat clau per resoldre la conjectura en el cas polinòmic en el pla afí complex. El beneficiari ha estudiat l’arbre valoratiu global de Favre i Jonsson i ha reinterpretat algunes nocions i resultats des d’un punt de vista més geomètric. Així mateix, ha estudiat la demostració de la conjectura de Bellon – Viallet en el cas polinòmic en el pla afí complex com a primer pas per trobar una demostració en el cas local i projectiu global en estudis futurs. El projecte inclou un estudi detallat de l'arbre valoratiu global des d'un punt de vista geomètric i els primers passos de la demostració de la conjectura de Bellon - Viallet en el cas polinòmic en el pla afí complex que van efectuar Favre i Jonsson.

Macaulay style formulas for sparse resultants

We present formulas for computing the resultant of sparse polyno- mials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials.

The Brauer group and the second Abel-Jacobi map for 0-cycles on algebraic varieties

This paper focuses on the connection between the Brauer group and the 0-cycles of an algebraic variety. We give an alternative construction of the second l-adic Abel-Jacobi map for such cycles, linked to the algebraic geometry of Severi-Brauer varieties on X. This allows us then to relate this Abel-Jacobi map to the standard pairing between 0-cycles and Brauer groups (see [M], [L]), completing results from [M] in this direction. Second, for surfaces, it allows us to present this map according to the more geometrical approach devised by M. Green in the framework of (arithmetic) mixed Hodge structures (see [G]). Needless to say, this paper owes much to the work of U. Jannsen and, especially, to his recently published older letter [J4] to B. Gross.

Extensions of maps defined on many fibres

Let S be a fibred surface. We prove that the existence of morphisms from non countably many fibres to curves implies, up to base change, the existence of a rational map from S to another surface fibred over the same base reflecting the properties of the original morphisms. Under some conditions of unicity base change is not needed and one recovers exactly the initial maps.

Bounds on multisecant lines

The purpose of this paper is two fold. First, we give an upper bound on the orderof a multisecant line to an integral arithmetically Cohen-Macaulay subscheme in Pn of codimension two in terms of the Hilbert function. Secondly, we givean explicit description of the singular locus of the blow up of an arbitrary local ring at a complete intersection ideal. This description is used to refine standardlinking theorem. These results are tied together by the construction of sharp examples for the bound, which uses the linking theorems.

On the motive of a quotient variety

We show that the motive of the quotient of a scheme by a finite group coincides with the invariant submotive.