Trigonal Gorenstein curves

We notice that the Maroni invariant of a trigonal Gorenstein curve of arithmetic genus g larger than four may be equal to zero, and we show that this happens if and only if the g₃¹ admits a non-removable base point, which is necessarily a singularity of the curve. We realize and study trigonal curves on rational scrolls, which in the case, where the g₃¹ admits a base point Q, degenerate to a cone with vertex Q.     

On the connectedness of the scheme of multisecants to a projective curve

Here we study the connectedness of the scheme of multisecant linear spaces to a curveC P ⁿ . In particular we prove the connectedness of the scheme of trisecant lines of a smoothC P ³ with non-special hyperplane section.     

On the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">d</Emphasis></Subscript>

Let Y be a smooth, connected, projective complex curve. In this paper, we study the Hurwitz spaces which parameterize branched coverings of Y whose monodromy group is a Weyl group of type D _d and whose local monodromies are all reflections except one. We prove the irreducibility of these spaces when and successively we extend the result to curves of genus g ≥ 1.     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

Abel maps and presentation schemes

We sharpen the two main tools used to treat the compactified Jacobian of a singular curve: Abel maps and presentation schemes. First we prove a smoothness theorem for bigraded Abel maps. Second we study the two complementary filtrations provided by the images of certain Abel maps and certain presentation schemes. Third we study a lifting of the Abel map of bidegree (m, 1) to the corresponding presentation scheme. Fourth we prove that, if a curve is blown up at a double point, then the corresponding presentation scheme is a ¹-bundle. Finally, using Abel maps of bidegree (m, 1), we characterize the curves having double points at worst.     

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.     

Versality of linearly normal non-special embeddings

LetX⊂P ⁿ be a smooth non-degenerate non special linearly normal projective curve. Here we classify all such embeddings ofX such that for every hyperplaneM ofP ⁿ the family of all hyperplane sections ofX is a versal deformation of the zerodimensional schemeX∩M.

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Sunto SiaX una curva liscia e proiettiva. Qui si classificano le immersioni non-speciali linearmente normali diX inP ⁿ tali che per ogni iperpianoM diP ⁿ la famiglia delle sezioni iperpiane diX induce una deformazione versale diX∩M.

     

Moduli spaces of principal <Emphasis Type="Italic">F</Emphasis>-bundles

In this paper we construct certain moduli spaces, which we call moduli spaces of (principal) F-bundles, and study their basic properties. These spaces are associated to triples consisting of a smooth projective geometrically connected curve over a finite field, a split reductive group G, and an irreducible algebraic representation .of of Our spaces generalize moduli spaces of F-sheaves, studied by Drinfeld and Lafforgue, which correspond to the case G = GL^r and is the tensor product of the standard representation and its dual. The importance of the moduli spaces of F-bundles is due to the belief that Langlands correspondence is realized in their cohomology.     

Smooth 4-folds which contain a P1-bundle¶as an ample divisor

We show that if a smooth projective 4-fold M contains an ample divisor A which is P ¹-bundle π :A→S over a smooth projective surface S, π is extended to a P ²-bundle π :S→S, unless $A$ is isomorphic to P ²×P ¹. Received: 28 September 1998 / Revised version: 16 August 1999     

Two dimensional scrolls contained in quadric cones in ℙ5

Smooth, complex, ruled surfaces embedded in ℙ⁵ as linearly normal scrolls, such that they are contained in a quadric cone, are considered. Rational scrolls and some elliptic scrolls are shown to be the only ones contained in cones of rank 5. Results on scrolls contained in cones of lower ranks are also obtained.     

The singular locus of the secant varieties of a smooth projective curve

Let X be a smooth irreducible non-degenerated projective curve in some projective space P^N. Let r be a positive integer such that 2r + 1 < N and let S_r(X) be the r-th secant variety of X. It is a variety of dimension 2r + 1. In this paper we prove that the singular locus is the (r - 1)-th secant variety S_{r- 1}(X) if X does not have any (2r + 2)-secant 2r-space divisor. Received: 26 November 2002     

Isolated rational curves on K3-fibered Calabi–Yau threefolds

In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙ ^{n 1}}×ℙ¹. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau variety in projective space. Received: 14 October 1997 / Revised version: 18 January 1998     

Vector bundles on ruled Fano 3-folds

Let be a ruled Fano 3-fold. The goal of this paper is to compute the dimension, prove the irreducibility and smoothness and describe the structure of the moduli space M _L (2;c ₁,c ₂) of L-stable, rank 2 vector bundles E on X with certain Chern classes and for a suitable polarization L closely related to c ₂. More precisely, we will cover the study of some moduli spaces M _L (2;c ₁,c ₂) such that the generic point is given as a non-trivial extension of line bundles. This work nicely reflects the general philosophy that moduli spaces inherits a lot of geometrical properties of the underlying variety. Received: 16 February 1999 / Revised version: 2 July 1999     

Q-curves of degree 5 and jacobian surfaces of GL2-type

We construct a parametric family {E ^(±)(s,t,u)} of minimal Q-curves of degree 5 over the quadratic fields Q , and the family {C(s,t,u)} of genus two curves over Q covering E ^{(+)(s,t,u) whose jacobians are abelian surfaces of GL₂-type. We also discuss the modularity for them and the sign change between E ^{(+)(s,t,u) and its twist E ⁽⁻⁾(s,t,u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C(s,t,u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces A _f attached to cusp forms of Neben type character of level N= 29, 229, 349, 461, and 509. Received: 23 September 1997 / Revised version: 26 May 1998