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We study and compare the webs $\boldsymbol{\mathcal W}_{{\rm dP}_d}$ defined by the conic fibrations on a given smooth del Pezzo surface ${\rm dP}_d$ of degree $d$ for $d=4$ and $d=5$. In a previous paper, we proved that for any positive $d\leq 6$, the web by conics $\boldsymbol{\mathcal W}_{{\rm dP}_d}$ carries a particular abelian relation ${\bf HLog}_d$, whose components all are weight $7-d$ antisymmetric hyperlogarithms. The web $\boldsymbol{\mathcal W}_{{\rm dP}_5}$ is a geometric model of the exceptional Bol's web and the relation ${\bf HLog}_5$ corresponds to the famous `Abel's identity' $(\boldsymbol{{\mathcal A}b})$ of the dilogarithm. Bol's web together with $(\boldsymbol{{\mathcal A}b})$ enjoy several remarkable properties of different kinds. We show that almost all of them admit natural generalizations to the pair $\big( \boldsymbol{\mathcal W}_{{\rm dP}_4}, {\bf HLog}_4\big)$., Comment: 100 pages

For any $d\in \{1,\ldots,6\}$, we prove that the web of conics on a del Pezzo surface of degree $d$ carries a functional identity whose components are antisymmetric hyperlogarithms of weight $7-d$. Our approach is uniform with respect to $d$ and relies on classical results about the action of the Weyl group on the set of lines on the del Pezzo surface. These hyperlogarithmic functional identities are natural generalizations of the classical 3-term and (Abel's) 5-term identities satisfied by the logarithm and the dilogarithm, which correspond to the cases when $d=6$ and $d=5$ respectively., Comment: Second version, two references and a few comments added, 20 pages, 1 figure

For $d$ ranging from 2 to 6, we prove that the web by conics naturally defined on any smooth del Pezzo surface of degree $d$ carries an interesting functional identity whose components all are a certain antisymmetric hyperlogarithm of weight $7-d$. Our approach is uniform with respect to $d$ and at the end relies on classical results about the action of Weyl groups on the set of lines contained in the considered del Pezzo surface. This series of `del Pezzo's hyperlogarithmic functional identities' is a natural generalization of the famous and well-know 3-term and 5-term identities of the logarithm and dilogarithm ('Abel's relation') which correspond to the cases when $d=6$ and $d=5$ respectively. This text ends with a section containing several questions and some possibly interesting perspectives., Comment: Long preliminary version, not intended to be published in this form in a peer reviewed journal. 49 pages, 2 figures

We discuss the curvilinear web $\boldsymbol{\mathcal W}_{0,n+3}$ on the moduli space $\mathcal M_{0,n+3}$ of projective configurations of $n+3$ points on $\mathbf P^1$ defined by the $n+3$ forgetful maps $\mathcal M_{0,n+3}\rightarrow \mathcal M_{0,n+2}$. We recall classical results which show that this web is linearizable when $n$ is odd, or is equivalent to a web by conics when $n$ is even. We then turn to the abelian relations (ARs) of these webs. After recalling the well-known case when $n=2$ (related to the 5-terms functional identity of the dilogarithm), we focus on the case of the 6-web $\boldsymbol{\mathcal W}_{{0,6}}$. We show that this web is isomorphic to the web formed by the lines contained in Segre's cubic primal $\boldsymbol{S}\subset \mathbf P^4$ and that a kind of `Abel's theorem' allows to describe the ARs of $\boldsymbol{\mathcal W}_{{0,6}}$ by means of the abelian 2-forms on the Fano surface $F_1(\boldsymbol{S})\subset G_1(\mathbf P^4)$ of lines contained in $\boldsymbol{S}$. We deduce from this that $\boldsymbol{\mathcal W}_{{0,6}}$ has maximal rank with all its ARs rational, and that these span a space which is an irreducible $\mathfrak S_6$-module. Then we take up an approach due to Damiano that we correct in the case when $n$ is odd: it leads to an abstract description of the space of ARs of $\boldsymbol{\mathcal W}_{0,n+3}$ as a $\mathfrak S_{n+3}$-representation. In particular, we obtain that this web has maximal rank for any $n\geq 2$. Finally, we consider `Euler's abelian relation $\boldsymbol{\mathcal E}_n$', a particular AR for $\boldsymbol{\mathcal W}_{0,n+3}$ constructed by Damiano from a characteristic class on the grassmannian of 2-planes in $\mathbf R^{n+3}$ by means of Gelfand-MacPherson theory of polylogarithmic forms. We give an explicit conjectural formula for the components of $\boldsymbol{\mathcal E}_n$ that we prove to be correct for $n\leq 12$., Comment: 121 pages, 4 figures. Second version : a few typos have been corrected

In this text, we investigate webs which can be associated to cluster algebras from the point of view of the abelian functional equations these webs carry, focusing on the polylogarithmic ones. We introduce a general notion of webs whose rank is `As Maximal as Possible' (AMP) and show that many webs associated either to polylogarithmic functional equations or to cluster algebras are of this type. In particular, we prove a few results and state some conjectures about cluster webs associated to (pairs of) Dynkin diagrams. Along the way, we show that many of the classical functional equations satisfied by low-order polylogarithms (such as Spence-Kummer's equation of the trilogarithm or the tetralogarithmic one of Kummer) are of cluster type., Comment: 298 pages, many figures. Preliminary version. Comments are welcome

In the genus one case, we make explicit some constructions of Veech on flat surfaces and generalize some geometric results of Thurston about moduli spaces of flat spheres as well as some equivalent ones but of an analytico-cohomological nature of Deligne-Mostow, which concern the monodromy of Appell-Lauricella hypergeometric functions. In the twin paper arXiv:1604.01812, we follow Thurston's approach and study moduli spaces of flat tori with conical singularities and prescribed holonomy by means of geometrical methods relying on surgeries for flat surfaces. In the present paper, we study the same objects making use of analytical and cohomological methods, more in the spirit of Deligne-Mostow's paper., Comment: 156 pages, 20 figures. Preliminary version. Comments are welcome

We generalise to the genus one case several results of Thurston concerning moduli spaces of flat Euclidean structures with conical singularities on the two dimensional sphere. More precisely, we study the moduli space of flat tori with $n$ cone points and a prescribed holonomy $\rho$. In his paper `Flat Surfaces' Veech has established that under some assumptions on the cone angles, such a moduli space ${\mathcal{F}}_{[\rho]}\subset \mathscr M_{1,n}$ carries a natural geometric structure modeled on the complex hyperbolic space ${\mathbb C}{\mathbb{H} }^{n-1}$ which is not metrically complete. Using surgeries for flat surfaces, we prove that the metric completion $\overline{{\mathcal{F}}_{[\rho]}}$ is obtained by adjoining to $ {\mathcal{F}}_{[\rho]} $ certain strata that are themselves moduli spaces of flat surfaces of genus 0 or 1, obtained as degenerations of the flat tori whose moduli space is $ {\mathcal{F}}_{[\rho]}$. We show that the ${\mathbb C}{\mathbb{H} }^{n-1}$-structure of $ {\mathcal{F}}_{[\rho]}$ extends to a complex hyperbolic cone-manifold structure of finite volume on $ \overline{{\mathcal{F}}_{[\rho]}}$ and we compute the cone angles associated to the different strata of codimension 1. Finally, we address the question of whether or not the holonomy of Veech's ${\mathbb C}{\mathbb{H} }^{n-1}$-structure on $ \mathcal F_\rho$ has a discrete image in $ {\rm Aut}({\mathbb C}{\mathbb{H} }^{n-1})=\mathrm{PU}(1,n-1)$. We outline a general strategy to find moduli spaces $\mathcal F_{[\rho]}$ whose ${\mathbb C}{\mathbb{H} }^{n-1}$-holonomy gives rise to lattices in $\mathrm{PU}(1,n-1)$ and eventually we give a finite list of $\mathcal F_{[\rho]}$'s whose holonomy is a complex hyperbolic arithmetic lattice., Comment: 86 pages, 16 figures, 2 tables. Revised version (details added, pictures improved). To appear in GAFA

In arXiv:1302.3142, it has been proved that for r>1, n>1 and d>(r+1)(n-1)+1, a d-web of type (r,n) with maximal rank is algebraizable in the classical sense, except maybe when n>2 and d = (r+2)(n-1)+1. In the present paper, one considers this particular case. Under these hypotheses on r, n and d, one constructs some examples of `exceptional algebraic webs': these are generalized algebraic webs of maximal rank that aren't algebraizable in the classical sense., Comment: 34 pages, in French, 1 table. To appear in Mathematische Annalen

We apply the results of arXiv:1109.3573 to study quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension four and five., Comment: 22 pages, 9 figures

As proved recently in [PT], for varieties $X^{r+1}\subset \mathbb P^N$ such that through $n\geq 2$ general points there passes an irreducible curve $C$ of degree $\delta\geq n-1$ we have $N\leq \pi(r,n,\delta+r(n-1)+2)$, where $\pi(r,n,d)$ is the Castelnuovo-Harris bound function for the geometric genus of an irreducible non-degenerate variety $Y^r\subset\mathbb P^{n+r-1}$ of degree $d$. A lot of examples of varieties as in the title and attaining the previous bound for the embedding dimension are constructed from Castelnuovo varieties and were thus dubbed {\it of Castelnuovo type} in [PT], where it is also proved that all extremal varieties as above are of this kind, except possibly when $n>2$, $r>1$ and $\delta=2n-3$. One of the main results of the paper is the classification of extremal varieties $X^{r+1}\subset \mathbb P^{2r+3}$ 3-covered by twisted cubics and not of Castelnuovo type. Interesting examples are provided by the so called {\it twisted cubics over complex Jordan algebras of rank 3}, as pointed out by Mukai. By relating to an extremal variety 3-covered by twisted cubics, via tangential projection, a quadro-quadric Cremona transformation in $\mathbb P^r$ we are able to classify all these object either for $r\leq 4$ or under the smoothness assumption. In the last case we obtain that they are either smooth rational normal scrolls (hence of Castelnuovo type) or the Segre embeddings of $\p^1\times Q^r$ or one of the four Lagrangian Grassmannians. We end by discussing some open problems pointing towards the equivalence of these apparently unrelated objects: extremal varieties 3-covered by twisted cubics, quadro-quadric Cremona transformations of $\mathbb P^r$ and complex Jordan algebras of dimension $r+1$ and of rank three., Comment: 30 pages; accepted for publication in Comm. Math. Helv. (november 2010)

For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and 3-covered by twisted cubic curves, up to projective equivalence; quadro-quadric Cremona transformations of $ \mathbb P^{n-1}$, up to linear equivalence; $n$-dimensional complex Jordan algebras of rank three, up to isotopy. We also provide some applications to the classification of particular classes of varieties in the class defined above and of quadro-quadric Cremona transformations, proving also a structure theorem for these birational maps and for varieties 3-covered by twisted cubics by reinterpreting for these objects the solvability of the radical of a Jordan algebra., Comment: 30 pages, 1 figure. Corrected typos

This book was intended to serve as supporting material for a mini-course on web geometry delivered at the 27th Brazilian Mathematical Colloquium which took place at IMPA in the last week of July 2009., Comment: 208 pages, 20 figures

Given integers r>1, n>1 and q> n-2, we consider projective varieties X of dimension r+1 such that through n generic points of X passes a rational curve of degree q, contained in X. More precisely, we study the class X_{r+1,n}(q) of such varieties which moreover generate a projective space of the maximal dimension. We determine all varieties of a class X_{r+1,n}(q) when q is not equal to 2n-3. In particuliar, we show that there exists a variety X' in P^{r+n-1}, of minimal degree and a birational map F: X'---> X which sends a generic section of X' by a P^{n-1} onto a rational normal curve of degree q. Without hypothesis on q, we define a quasi-grassmannian structure on the space of the rational normal curves of degree q contained in a variety X of the class X_{r+1,n}(q). We prove that X is of the form described above if and only if this quasi-grassmannian structure is flat. We also give examples of varieties of the classes X_{r+1,3}(3) et X_{r+1,4}(5) which are not of this form., Comment: 50 pages, in French

We give a simple analytic criterion which characterizes linearizable 1-codimensional webs. Then we give an invariant geometrical interpretation of it, in term of projective connection. We explain then how our approach allows to study linearization of more general objects than 1-codimensional webs. By way of illustration, we treat some explicit interesting examples., Comment: spelling mistakes corrected

Codimension one webs are configurations of finitely many codimension one foliations in general position. Much of the classical theory evolved around the concept of abelian relation: a functional relation among the first integrals of the foliations defining the web reminiscent of Abel's addition theorem in classical algebraic geometry. The abelian relations of a given web form a finite dimensional vector space with dimension (the rank of the web) bounded by Castelnuovo number p(n,k) where n is the dimension of the ambient space and k is the number of foliations defining the web. A fundamental problem in web geometry is the classification of exceptional webs, that is, webs of maximal rank not equivalent to the dual of a projective curve. Recently, J.-M. Trepreau proved that there are no exceptional k-webs for n>2 and k > 2n-1. In dimension two there are examples of exceptional k-webs for arbitrary k and the classification problem is wide open. In this paper, we classify the exceptional Completely Decomposable Quasi-Linear (CDQL) webs globally defined on compact complex surfaces. By definition, the CDQL (k+1)-webs are formed by the superposition of k linear foliations and one non-linear foliation. For instance, we show that up to projective transformations there are exactly four countable families and thirteen sporadic exceptional CDQL webs on the projective plane., Comment: 75 pages, 7 figures

We investigate the space of abelian relations of planar webs admitting infinitesimal automorphisms. As an application, we construct 4k-14 new algebraic families of global exceptionnal k-webs on the projective plane, for each k >4., Comment: 8 pages

We give a one parameter family of exceptional planar 5-webs. Each web is formed by four pencils of lines and by a foliation defined by the level curves of a function sn_k(x)sn_k(y) where sn_k denotes a Jacobi's elliptic function.

In this paper, we study local solutions F=(F1,..,Fn) of a general functional equation of the form F1(U1(x,y))+....+Fn(Un(x,y))=0. A such equation will be called an ``abelian functional equation'' (Afe). We will restrict ourselves to the case when the inner functions Ui's are real rational functions. First we prove that if the components Fi of F are measurable then they are analytic and are characterized as solutions of a certain linear differential equation constructed from the Ui's. Then we apply our results to the resolution of Afe's which are generalised versions of the classical functionals equations satisfied by low order polylogarithms. In the last section, these results, interpreted in the frame of web geometry, give us new exceptional webs, related to configurations of points in CP2. Finally, we study the problem of characterizing the dilogarithm (resp. the trilogarithm) by Rogers (resp. Spence-kummer) functional equation., Comment: 30 pages, 4 figures