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For a complex elliptic curve $E$ and a point $p$ of order $n$ on it, the images of the points $p_k=kp$ under the Weierstrass embedding of $E$ into $\mathbb{C}\mathbb{P}^2$ are collinear if and only if the sum of indices is divisible by $n$. Thus, it provides a realization of a certain matroid. We study this matroid in detail and prove that its realization space is isomorphic (over $\mathbb{C}$) to the modular curve $X_1(n)$, provided $n\geq 10$, which also provides an integral model of $X_1(n)$. In the process, we find a connection to the classical Ceva and B\"or\"oczky examples of special point and line configurations. We also discuss the situation for smaller values of $n$., Comment: 25 pages, two ancillary files

We construct new examples of free curve arrangements in the complex projective plane using point-line operators recently defined by the second author. In particular, we construct a new example of a conic-line arrangement with ordinary quasi-homogeneous singularities that has non-trivial monodromy., Comment: 12 pages, one ancillary file

This paper further studies the matroid realization space of a specific deformation of the regular $n$-gon with its lines of symmetry. Recently, we obtained that these particular realization spaces are birational to the elliptic modular surfaces $\Xi_{1}(n)$ over the modular curve $X_1(n)$. Here, we focus on the peculiar cases when $n=7,8$ in more detail. We obtain concrete quartic surfaces in $\mathbb{P}^3$ equipped with a dominant rational self-map stemming from an operator on line arrangements, which yields K3 surfaces with a dynamical system that is semi-conjugated to the plane., Comment: 16 pages, 2 figures, 1 ancillary file

We investigate the matroid realization space of a specific deformation of the regular $n$-gon with its lines of symmetry. It turns out that these particular realization spaces are birational to the elliptic modular surfaces $\Xi_{1}(n)$ over the modular curve $X_1(n)$. We obtain in that way a model of $\Xi_{1}(n)$ defined over the rational numbers. Furthermore, a natural geometric operator acts on these matroid realizations. On the elliptic modular surface this operator corresponds to the multiplication by $-2$ on the elliptic curves. That gives a new geometric way to compute the multiplication by $-2$ on elliptic curves., Comment: 15 pages, 4 figures. Comments welcome

We study some natural operators acting on configurations of points and lines in the plane and remark that many interesting configurations are fixed points for these operators. We review ancient and recent results on line or point arrangements though the realm of these operators. We study the first dynamical properties of the iteration of these operators on some line arrangements., Comment: 33 pages; improved version, references updated and some typos fixed thanks to some comments, a remark added (Remark 38)

The operator $\Lambda_{\{2\},\{3\}}$ acting on line arrangements is defined by associating to a line arrangement \mathcal{A}, the line arrangement which is the union of the lines containing exactly three points among the double points of \mathcal{A}. We say that six lines not tangent to a conic form an unassuming arrangement if the singularities of their union are only double points, but the dual line arrangement has six triple points, six 5-points and 27 double points. The moduli space of unassuming arrangements is the union of a point and a line. The image by the operator $\Lambda_{\{2\},\{3\}}$ of an unassuming arrangement is again an unassuming arrangement. We study the dynamics of the operator $\Lambda_{\{2\},\{3\}}$ on these arrangements and we obtain that the periodic arrangements are related to the Ceva arrangements of lines., Comment: 21 pages, ancillary file with magma code used added

We prove that every K3 surface with automorphism group $(\mathbb{Z}/2\mathbb{Z})^2$ admits an explicit birational model as a double sextic surface. This model is canonical for Picard number greater than 10. For Picard number greater than 9, the K3 surfaces in question possess a second birational model, in the form of a projective quartic hypersurface, generalizing the Inose quartic., Comment: 28 pages, 10 figures

Let $X$ be a compact, complex surface of general type whose cotangent bundle $\Omega_X$ is strongly semi-ample. We study the pluri-cotangent maps of $X$, namely the morphisms $\psi_n \colon \mathbb{P}(\Omega_X) \to \mathbb{P}(H^0(X, \, S^n \Omega_X))$ defined by the vector space of global sections $H^0(X, \, S^n \Omega_X)$., Comment: 25 pages. Final version, to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

A generalized Kummer surface $X$ of order $3$ is the minimal resolution of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism. We study a generalization of a problem of Shioda for classical Kummer surfaces, which is to understand how much $X$ is determined by $A$ and conversely. The surface $X$ posses a big and nef divisor $L_{X}$ such that $L_{X}^{2}=0$ or $2$ mod $6$. We show that for surfaces with $L_{X}^{2}=6k$ with $k\neq0,6\,mod\,9$, the surface $X$ determines the transcendental lattice $T(A)$ of $A$ and the Hodge structure on $T(A)$. Conversely if $A$ and $B$ are Fourier-Mukai partners (i.e. if the Hodge structures of their transcendental lattices are isomorphic) and $Y$ is the generalized Kummer surface which is the minimal resolution of the quotient of $B$ by an order $3$ symplectic automorphism, we obtain that $X$ and $Y$ are isomorphic. These results are also know to hold for surfaces with $L_{X}^{2}=2\,mod\,6$ from a previous work. When $k=0\text{ or }6\,mod\,9,$ we show that $X$ determines $T(A)$ and its Hodge structure, but the converse does not hold in general., Comment: 21 pages, comments welcome

We study generalized Kummer surfaces Km$_{3}(A)$, by which we mean the K3 surfaces obtained by desingularization of the quotient of an abelian surface $A$ by an order $3$ symplectic automorphism group. Such a surface carries $9$ disjoint configurations of two smooth rational curves $C,C'$ with $CC'=1$. This $9{\bf A}_{2}$-configuration plays a role similar to the Nikulin configuration of $16$ disjoint smooth rational curves on (classical) Kummer surfaces. We study the (generalized) question of T. Shioda: suppose that Km$_{3}(A)$ is isomorphic to Km$_{3}(B)$, does that imply that $A$ and $B$ are isomorphic? We answer by the negative in general, by two methods: by a link between that problem and Fourier-Mukai partners of $A$, and by construction of $9{\bf A}_{2}$-configurations on Km$_{3}(A)$ which cannot be exchanged under the automorphism group., Comment: 27 pages ; version with more details and clarity in the proofs of the results

Given coprime positive integers $d',d''$, B\'ezout's Lemma tells us that there are integers $u,v$ so that $d'u-d''v=1$. We show that, interchanging $d'$ and $d''$ if necessary, we may choose $u$ and $v$ to be Loeschian numbers, i.e., of the form $|\alpha|^2$, where $\alpha\in\mathbb{Z}[j]$, the ring of integers of the number field $\mathbb{Q}(j)$, where $j^2+j+1=0$. We do this by using Atkin-Lehner elements in some quaternion algebras $\mathcal{H}$. We use this fact to count the number of conjugacy classes of elements of order 3 in an order $\mathcal{O}\subset\mathcal{H}$., Comment: 20 pages, comments welcome

A generalized Kummer surface $X=Km_{3}(A,G_{A})$ is the minimal resolution of the quotient of a $2$-dimensional complex torus by an order 3 symplectic automorphism group $G_{A}$. A Kummer structure on $X$ is an isomorphism class of pairs $(B,G_{B})$ such that $X\simeq Km_{3}(B,G_{B})$. When the surface is algebraic, we obtain that the number of Kummer structures is linked with the number of order $3$ elliptic points on some Shimura curve naturally related to $A$. For each $n\in\mathbb{N}$, we obtain generalized Kummer surfaces $X_{n}$ for which the number of Kummer structures is $2^{n}$. We then give a classification of the moduli spaces of generalized Kummer surfaces. When the surface is non algebraic, there is only one Kummer structure, but the number of irreducible components of the moduli spaces of such surfaces is large compared to the algebraic case. The endomorphism rings of the complex $2$-tori we study are mainly quaternion orders, these order contain the ring of Eisenstein integers. One can also see this paper as a study of quaternion orders $\mathcal{O}$ over $\mathbb{Q}$ that contain the ring of Eisenstein integers. We obtain that such order is determined up to isomorphism by its discriminant, and when the quaternion algebra is indefinite, the order $\mathcal{O}$ is principal., Comment: 33 pages (short version), Final version

In this paper we study the geometry of the $14$ families of K3 surfaces of Picard number four with finite automorphism group, whose N\'eron-Severi lattices have been classified by \`E.B. Vinberg. We provide projective models, we identify the degrees of a generating set of the Cox ring and in some cases we prove the unirationality of the associated moduli space., Comment: 40 pages, 7 tables, 2 ancilliary files

A generalized Kummer surface $X$ obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $9\mathbf{A}_{2}$-configuration of $(-2)$-curves. Such a configuration plays the role of the $16\mathbf{A}_{1}$-configurations for usual Kummer surfaces. In this paper we construct $9$ other such $9\mathbf{A}_{2}$-configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $9\mathbf{A}_{2}$-configurations are obtained by taking the pullback of a certain configuration of $12$ conics which are in special position with respect to the branch curve, plus some singular quartic curves. We then construct some automorphisms of the K3 surface sending one configuration to another. We also give various models of $X$ and of the generic fiber of its natural elliptic pencil., Comment: 29 pages; revised version according to the referee's remarks

We study the geometry of the K3 surfaces $X$ with a finite number automorphisms and Picard number $\geq 3$. We describe these surfaces classified by Nikulin and Vinberg as double covers of simpler surfaces or embedded in a projective space. We study moreover the configurations of their finite set of $(-2)$-curves., Comment: 95 pages, 80 figures, 1 Table

We construct special conics configurations from some points configurations which are the singularities of the dual of a quartic curve., Comment: 13 pages, few corrections

Nikulin and Vinberg proved that there are only a finite number of lattices of rank $\geq 3$ that are the N\'eron-Severi group of projective K3 surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such K3 surfaces $X$, when these surfaces have moreover no elliptic fibrations. In that case we show that such K3 surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study the converse i.e. if the geometric description we obtained characterizes these surfaces. In $4$ cases the description is sufficient, in each of the $4$ other cases there is exactly another one possibility which we study. We obtain that at least 5 moduli spaces of K3 surfaces (among the 8 we study) are unirational., Comment: 32 pages

A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to results of Nikulin, the existence of a Nikulin configuration means that the K3 surface is a Kummer surface, moreover the abelian surface from the Kummer structure is determined by the $16$ curves. A classical question of Shioda is about the existence of non isomorphic Kummer structures on the same Kummer K3 surface. The question was studied by several authors, and it was shown that the number of non-isomorphic Kummer structures is finite, but no explicit geometric construction of such structures was given. In a previous paper, we constructed explicitly non isomorphic Kummer structures on some Kummer surfaces. In this paper we generalise the construction to Kummer surfaces with a weaker restriction on the degree of the polarization and we describe some cases where the previous construction does not work., Comment: 21 pages, revised version according to the referee's suggestions

We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify $X$ as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equation of the mirror $M$ of the orbifold ball quotient $(X,M)$ and by taking the quotient by an involution, we obtain an orbifold ball quotient surface with mirror birational to an interesting configuration of plane curves of degrees $1,2$ and $3$. We also exhibit an arrangement of four conics in the plane which provides the above-mentioned ball quotient orbifold surfaces.

We construct and study curves with low H-constants on abelian and K3 surfaces. Using the Kummer $(16_{6})$-configurations on Jacobian surfaces and some $(16_{10})$-configurations of curves on $(1,3)$-polarized Abelian surfaces, we obtain examples of configurations of curves of genus $>1$ on a generic Jacobian K3 surface with H-constants $<-4$., Comment: Final version, 12 pages

A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration $\mathcal{C}$, then $X$ is a Kummer surface $X=Km(B)$ where $B$ is an Abelian surface determined by $\mathcal{C}$. Let $B$ be a generic Abelian surface having a polarization $M$ with $M^{2}=k(k+1)$ (for $k>0$ an integer) and let $X=Km(B)$ be the associated Kummer surface. To the natural Nikulin configuration $\mathcal{C}$ on $X=Km(B)$, we associate another Nikulin configuration $\mathcal{C}'$; we denote by $B'$ the Abelian surface associated to $\mathcal{C}'$, so that we have also $X=Km(B')$. For $k\geq2$ we prove that $B$ and $B'$ are not isomorphic. We then construct an infinite order automorphism of the Kummer surface $X$ that occurs naturally from our situation. Associated to the two Nikulin configurations $\mathcal{C},$ $\mathcal{C}'$, there exists a natural bi-double cover $S\to X$, which is a surface of general type. We study this surface which is a Lagrangian surface in the sense of Bogomolov-Tschinkel, and for $k=2$ is a Schoen surface., Comment: 22 pages, refereed version

A generalized Kummer surface $X=Km(T,G)$ is the resolution of a quotient of a torus $T$ by a finite group of symplectic automorphisms $G$. We complete the classification of generalized Kummer surfaces by studying the two last groups which have not been yet studied. For these surfaces, we compute the associated Kummer lattice $K_{G}$, which is the minimal primitive sub-lattice containing the exceptional curves of the resolution $X\to T/G$. We then prove that a K3 surface is a generalised Kummer surface of type $Km(T,G)$ if and only if its N\'eron-Severi group contains $K_{G}$. For smooth-orbifold surfaces $\mathcal{X}$ of Kodaira dimension $\geq 0$, Kobayashi proved the orbifold Bogomolov Miyaoka Yau inequality $c_{1}^{2}(\mathcal{X})\leq3c_{2}(\mathcal{X}).$ For Kodaira dimension $2$, the case of equality is characterised as $\mathcal{X}$ being uniformized by the complex $2$-ball $\mathbb{B}_{2}$. For smooth-orbifold K3 and Enriques surfaces we characterize the case of equality as being uniformized by $\mathbb{C}^{2}$., Comment: 17 pages, Final version, Comments still welcome

An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo $p$. It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of $\mathbb F_3$. As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over $\mathbb F_q$ which attains Perret's and Weil's upper bounds., Comment: 12 pages

We construct two complex-conjugated rigid surfaces with $p_g=q=2$ and $K^2=8$ whose universal cover is not biholomorphic to the bidisk. We show that these are the unique surfaces with these invariants and Albanese map of degree $2$, apart the family of product-quotient surfaces constructed by Penegini. This completes the classification of surfaces with $p_g=q=2, K^2=8$ and Albanese map of degree $2$., Comment: Final version. To appear in IMRN

We give an explicit construction for the $4$-dimensional family of Schoen surfaces by computing equations for their canonical images, which are $40$-nodal complete intersections of a quadric and the Igusa quartic in $\mathbb P^4$. We then study a particularly interesting example, with $240$ automorphisms and maximal Picard number., Comment: Final version

The Bounded Negativity Conjecture predicts that for every complex projective surface $X$ there exists a number $b(X)$ such that $C^2\geq -b(X)$ holds for all reduced curves $C\subset X$. For birational surfaces $f:Y\to X$ there have been introduced certain invariants (Harbourne constants) relating to the effect the numbers $b(X)$, $b(Y)$ and the complexity of the map $f$. These invariants have been studied previously when $f$ is the blowup of all singular points of an arrangement of lines in ${\mathbb P}^2$, of conics and of cubics. In the present note we extend these considerations to blowups of ${\mathbb P}^2$ at singular points of arrangements of curves of arbitrary degree $d$. We also considerably generalize and modify the approach witnessed so far and study transversal arrangements of sufficiently positive curves on arbitrary surfaces with the non-negative Kodaira dimension., Comment: This is the final version, incorporating the suggestions of the referee, to appear in Annales de l'Institut Fourier Grenoble

We show that smooth cubic hypersurfaces of dimension $n$ defined over a finite field ${\bf F}_q$ contain a line defined over ${\bf F}_q$ in each of the following cases: - $n=3$ and $q\ge 11$; - $n=4$ and $q\ne 3$; - $n\ge 5$. For a smooth cubic threefold $X$, the variety of lines contained in $X$ is a smooth projective surface $F(X)$ for which the Tate conjecture holds, and we obtain information about the Picard number of $F(X)$ and its 5-dimensional principally polarized Albanese variety $A(F(X))$., Comment: The example in Section 5.4.2 was corrected by Nicolas Addington. Kiran Kedlaya corrected an error in our proof of Theorem 5.2. He also kindly provided Proposition 5.5 and its proof. Daniel Bragg corrected an error in the statement of Theorem 4.12. Typo corrected in (5). This is a slightly expanded (and corrected) version of the text published in the Simons Publication Series

Similarly to the linear Harbourne constant recently defined, we study the elliptic $H$-constants of $\mathbb{P}^{2}$ and Abelian surfaces. We exhibit configurations of smooth plane cubic curves whose Harbourne index is arbitrarily close to $-4$. As a Corollary, we obtain that the global $H$-constant of any surface $X$ is less or equal to $-4$. Related to these problems, we moreover give a new inequality for the number and multiplicities of singularities of elliptic curves arrangements on Abelian surfaces, inequality which has a close similarity to the one of Hirzebruch for arrangements of lines in the plane., Comment: Revised and shorter version

We study a surface discovered by Stover which is the surface with minimal Euler number and maximal automorphism group among smooth arithmetic ball quotient surfaces. We study the natural map $\wedge^{2}H^{1}(S,\mathbb{C})\to H^{2}(S,\mathbb{C})$ and we discuss the problem related to the so-called Lagrangian surfaces. We obtain that this surface $S$ has maximal Picard number and has no higher genus fibrations. We compute that its Albanese variety $A$ is isomorphic to $(\mathbb{C}/\mathbb{Z}[\alpha])^{7}$, for $\alpha=e^{2i\pi/3}$., Comment: 7 pages, Comments welcome

The \emph{canonical degree} of a curve $C$ on a surface $X$ is $K_X\cdot C$. Our main result, is that on a surface of general type there are only finitely many curves with negative self--intersection and sufficiently large canonical degree. Our proof strongly relies on results by Miyaoka. We extend our result both to surfaces not of general type and to non--negative curves, and give applications, e.g. to finiteness of negative curves on a general blow--up of $\mathbb P^ 2$ at $n\geq 10$ general points (a result related to \emph{Nagata's Conjecture}). We finally discuss a conjecture by Vojta concerning the asymptotic behaviour of the ratio between the canonical degree and the geometric genus of a curve varying on a surface. The results in this paper go in the direction of understanding the \emph{bounded negativity} problem., Comment: 7 pages, 1 figure, comments welcome

Quaternionic Shimura surfaces are quotient of the bidisc by an irreducible cocompact arithmetic group. In the present paper we are interested in (smooth) quaternionic Shimura surfaces admitting an automorphism with one dimensional fixed locus; such automorphisms are involutions. We propose a new construction of surfaces of general type with $q=p_{g}=0$ as quotients of quaternionic Shimura surfaces by such involutions. These quotients have finite fundamental group., Comment: 22 pages ; comments welcome

We prove that for any number $r$ in $[2,3]$, there are spin (resp. non-spin minimal) simply connected complex surfaces of general type $X$ with $c_1^2(X)/c_2(X)$ arbitrarily close to $r$. In particular, this shows the existence of simply connected surfaces of general type arbitrarily close to the Bogomolov-Miyaoka-Yau line. In addition, we prove that for any $r \in [1,3]$ and any integer $q\geq 0$, there are minimal complex surfaces of general type $X$ with $c_1^2(X)/c_2(X)$ arbitrarily close to $r$, and $\pi_1(X)$ isomorphic to the fundamental group of a compact Riemann surface of genus $q$., Comment: 20 pages. Final version

We give an algorithm to compute the zeta function of the Fano surface of lines of a smooth cubic threefold $F$ into $\mathbb{P}^4$ defined over a finite field. We obtain some examples of Fano surfaces with supersingular reduction., Comment: Proof of Corollary 6 completed

Surfaces of general type with positive second Segre number are known to have big cotangent bundle. We give a new criterion ensuring that a surface of general type with canonical singularities has a minimal resolution with big cotangent bundle. This provides many examples of surfaces with negative second Segre number and big cotangent bundle., Comment: 11 pages. Comments welcome

We give a new construction of the irregular, generalized Lagrangian, surfaces of general type with p_g=5, \chi=2, K^2=8, recently discovered by Chad Schoen. Our approach proves that, if S is a general Schoen surface, its canonical map is a finite morphism of degree 2 onto a canonical surface with invariants p_g=5, \chi=6, K^2=8, a complete intersection of a quadric and a quartic hypersurface in P^4, with 40 even nodes., Comment: 10 pages

It has been proved by Adler that there exists a unique cubic hypersurface X in P^8 which is invariant under the action of the simple group PSL(2,19). In the present note we study the intermediate Jacobian of X and in particular we prove that the subjacent 85-dimensional torus is an Abelian variety. The symmetry group G=PSL(2,19) defines uniquely a G-invariant abelian 9-fold A(X), which we study in detail and describe its period lattice., Comment: 14 pages, comments welcome

A Fano surface of a smooth cubic threefold X in P^4 parametrizes the lines on X. In this note, we prove that a Fano surface satisfies the Tate conjecture over a field of finite type over the prime field and characteristic not 2., Comment: 4 pages ; the first version of this paper has been split into two parts

Surfaces of general type with positive second Segre number $s_2:=c_1^2-c_2>0$ are known by results of Bogomolov to be quasi-hyperbolic i.e. with finitely many rational and elliptic curves. These results were extended by McQuillan in his proof of the Green-Griffiths conjecture for entire curves on such surfaces. In this work, we study hyperbolic properties of minimal surfaces of general type with minimal $c_1^2$, known as Horikawa surfaces. In principle these surfaces should be the most difficult case for the above conjecture as illustrate the quintic surfaces in $\bP^3$. Using orbifold techniques, we exhibit infinitely many irreducible components of the moduli of Horikawa surfaces whose very generic member has no rational curves or even is algebraically hyperbolic. Moreover, we construct explicit examples of algebraically hyperbolic and (quasi-)hyperbolic orbifold Horikawa surfaces., Comment: 25 pages; final version to appear in J. Lond. Math. Soc

A fake quadric is a smooth minimal surface of general type with the same invariants as the quadric in P^3, i.e. K^2=2c_2=8 and q=p_g=0. We study here quaternionic fake quadrics i.e. fake quadrics constructed arithmetically by using some quaternion algebras over real number fields. We provide examples of quaternionic fake quadrics X with a non-trivial automorphism group and compute the invariants of the minimal desingularisation of the quotient of X by this group. In that way we obtain minimal surfaces of general type Z with q=p_g=0 and K^2=4,2 which contain the maximal number of disjoint nodal curves. We then prove that if a surface of general type has the same invariant as Z and same number of nodal curves, we can construct geometrically a surface of general type with K^2=2c_2=8., Comment: 22 pages, acknowledgements completed

Fano surfaces parametrize the lines of smooth cubic threefolds. In this paper, we study their quotients by some of their automorphism sub-groups. We obtain in that way some interesting surfaces of general type., Comment: 21 pages, corrections according to the referee request, to appear in Rendiconti Lincei

We study the configurations of genus 2 curves on the Fano surfaces of cubic threefolds. We establish a link between some involutive automorphisms acting on such a surface S and genus 2 curves on S. We give a partial classification of the Fano surfaces according to the automorphism group generated by these involutions and determine the configurations of their genus 2 curves. We study the Fano surface of the Klein cubic threefold for which the 55 genus 2 curves generate a rank 25=h^{1,1} index 2 subgroup of the N\'eron-Severi group., Comment: 15 pages

We study the Fano surface S of the Fermat cubic threefold. We prove that S is a degree 81 abelian cover of the degree 5 del Pezzo surface and that the complement of the union of 12 disjoint elliptic curves on S is a ball quotient. The lattice of this ball quotient is related to the Deligne-Mostow lattice number 1., Comment: 8 pages, extended and final version, to appear in the Proc. of. A.M.S

We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order 11. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index 2 sub-group of the N\'eron-Severi group and we obtain a set of generators of this group. The N\'eron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$., Comment: 15 pages

Curves of low genus on a surface carry important informations on that surface. We study the Fano surfaces of lines of cubic threefolds that contain 12 or 30 elliptic curves. We determine their Picard number and compute a basis of the N\'eron-Severi group of the Fano surface of the Fermat cubic threefold., Comment: 17 pages, minor changes, to be published in Michigan Math. J

We study surfaces of general type $S$ whose cotangent sheaf is generated by its global sections. We define a map called the cotangent map of $S$ that enables us to understand the obstructions to the ampleness of the cotangent sheaf of $S$. These obstructions are curves on $S$ that we call "non-ample". We classify surfaces with an infinite number of non-ample curves and we partly classify the non-ample curves., Comment: 21 pages

The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces., Comment: 17 pages, accepted and shortened version, the rest will appear in "Fano surfaces with 12 or 30 elliptic curves"