On a question of Sean Keel

Let X be a smooth projective surface defined over , and let L be a line bundle over X such that for every complete curve Y contained in X. A question of Keel asks whether L is ample. If X is a P¹-bundle over a curve, we prove that this question has an affirmative answer.     

Separators of points on algebraic surfaces

For a finite set of points X⊆Pⁿ and for a given point P∈X, the notion of a separator of P in X (a hypersurface containing all the points in X except P) and of the degree of P in X, (the minimum degree of these separators) has been largely studied. In this paper we extend these notions to a set of points X on a projectively normal surface S⊆Pⁿ, considering as separators arithmetically Cohen-Macaulay curves and generalizing the case S=P² in a natural way. We denote the minimum degree of such curves as and we study its relation to . We prove that if S is a variety of minimal degree these two terms are explicitly related by a formula, whereas only an inequality holds for other kinds of surfaces.     

Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f^∗O_P¹(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L.     

The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 2 and 3

Let X be a smooth projective curve of genus g?2 defined over an algebraically closed field k of characteristic p>0. Let M_X(r) be the moduli space of semi-stable vector bundles with fixed trivial determinant. The relative Frobenius map induces by pull-back a rational map . We determine the equations of V in the following two cases (1) (g,r,p)=(2,2,2) and X nonordinary with Hasse-Witt invariant equal to 1 (see math.AG/0005044 for the case X ordinary), and (2) (g,r,p)=(2,2,3). We also show the existence of base points of V, i.e., semi-stable bundles E such that F∗E is not semi-stable, for any triple (g,r,p).     

Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n.     

Ranges of densely defined generalized pseudomonotone perturbations of maximal monotone operators

Let X be a real reflexive Banach space and be maximal monotone. Let be quasibounded, finitely continuous and generalized pseudomonotone with X′⊂D(B), where X′ is a dense subspace of X such that X′∩D(A)≠∅. Let S⊂X∗. Conditions are given under which and intS⊂intR(A+B). Results of Browder concerning everywhere defined continuous and bounded operators B are improved. Extensions of this theory are also given using the degree theory of the last two authors concerning densely defined perturbations of nonlinear maximal monotone operators which satisfy a generalized (S₊)-condition. Applications of this extended theory are given involving nonlinear parabolic problems on cylindrical domains.     

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” we introduce are extensions by groups of analytic types of the usual extension groups attached to OX-modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X - and we especially consider the case when X is an “arithmetic curve”, namely the spectrum SpecOK of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers.Namely, for any two hermitian vector bundles and over X:=SpecOK, we attach a logarithmic size to any element α of , and we give an upper bound on in terms of slope invariants of and . We further illustrate this notion by relating the sizes of restrictions to points in P¹(Z) of the universal extension over to the geometry of PSL₂(Z) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K^′): when the base field K is Q, we establish that the size, which cannot increase under base change, is actually invariant when the field K^′ is an abelian extension of K, or when is a direct sum of root lattices and of lattices of Voronoi's first kind.The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.     

On unramified normal coverings of hyperelliptic curves

It is well known that the number of unramified normal coverings of an irreducible complex algebraic curve C with a group of covering transformations isomorphic to Z₂⊕Z₂ is (2^4g−3⋅2^2g+2)/6. Assume that C is hyperelliptic, say . Horiouchi has given the explicit algebraic equations of the subset of those covers which turn out to be hyperelliptic themselves. There are of this particular type. In this article, we provide algebraic equations for the remaining ones.     

The compactified Picard scheme of the compactified Jacobian

Let C be an integral projective curve in any characteristic. Given an invertible sheaf L on C of degree 1, form the corresponding Abel map , which maps C into its compactified Jacobian, and form its pullback map , which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, double points, then is known to be an isomorphism. We prove that always extends to a map between the natural compactifications, , and that the extended map is an isomorphism if C has, at worst, ordinary nodes and cusps.     

On varieties of almost minimal degree III: Tangent spaces and embedding scrolls

Let X⊂P^r be a variety of almost minimal degree which is the projected image of a rational normal scroll from a point p outside of . In this paper we study the tangent spaces at singular points of X and the geometry of the embedding scrolls of X, i.e. the rational normal scrolls Y⊂P^r which contain X as a codimension one subvariety.     

Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.     

Independence of ? in Lafforgue's theorem

Let X be a smooth curve over a finite field of characteristic p, let ?≠p be a prime number, and let be an irreducible lisse -sheaf on X whose determinant is of finite order. By a theorem of L. Lafforgue, for each prime number ?′≠p, there exists an irreducible lisse -sheaf on X which is compatible with , in the sense that at every closed point x of X, the characteristic polynomials of Frobenius at x for and are equal. We prove an “independence of ?” assertion on the fields of definition of these irreducible ?′-adic sheaves : namely, that there exists a number field F such that for any prime number ?′≠p, the -sheaf above is defined over the completion of F at one of its ?′-adic places.     

Adjoint divisors on algebraic curves

Let C be a numerically connected curve lying on a smooth algebraic surface. We show that if is an ample invertible sheaf satisfying some technical numerical hypotheses then is normally generated. As a corollary we show that the sheaf ω_C^⊗2 on a numerically connected curve C of arithmetic genus p_a?3 is normally generated if ω_C is ample and does not exist a subcurve B⊂C such that p_a(B)=1=B(C−B).     

Hilbert curves of polarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V_X associated to the Hilbert polynomial χ(x₁L₁+?+x_ρL_ρ), where L₁,…,L_ρ is a basis of , ρ being the Picard number of X, and x₁,…,x_ρ are complex variables. After studying general properties of V_X we specialize to the Hilbert curve of a polarized variety (X,L), namely the plane curve of degree dim(X) associated to χ(xK_X+yL). Special emphasis is given to the case of polarized threefolds.     

Families of low dimensional determinantal schemes

A scheme X⊂Pⁿ of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t×t minors of a homogeneous t×(t+c−1) matrix (f_ij). Given integers a₀≤a₁≤?≤a_t+c−2 and b₁≤?≤b_t, we denote by the stratum of standard determinantal schemes where f_ij are homogeneous polynomials of degrees a_j−b_i and is the Hilbert scheme (if n−c>0, resp. the postulation Hilbert scheme if n−c=0).Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of in and we show that is generically smooth along under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of appearing in Kleppe and Miró-Roig (2005) [25].     

Curves having one place at infinity and linear systems on rational surfaces

Denoting by L_d(m₀,m₁,…,m_r) the linear system of plane curves of degree d passing through r+1 generic points p₀,p₁,…,p_r of the projective plane with multiplicity m_i (or larger) at each p_i, we prove the Harbourne-Hirschowitz Conjecture for linear systems L_d(m₀,m₁,…,m_r) determined by a wide family of systems of multiplicities and arbitrary degree d. Moreover, we provide an algorithm for computing a bound for the regularity of an arbitrary system , and we give its exact value when is in the above family. To do that, we prove an H¹-vanishing theorem for line bundles on surfaces associated with some pencils “at infinity”.