Double rational normal curves with linear syzygies

In this note we are looking after nilpotent projective curves without embedded points, which have rational normal curves of degree d as support, are defined (scheme-theoretically) by quadratic equations, have degree 2d and have only linear syzygies. We show that, as expected, no such curve does exist in ℙ ^d , and then consider doublings in a bigger ambient space. The simplest and trivial example is that of a double line in the plane. We show that the only possibility is to take rational normal curves in ℙ ^d embedded further in ℙ^{2 d} and to take a certain doubling in the sense of Ferrand (cf. [5]) in ℙ^{2 d} . These double curves have the Hilbert polynomial H(t)=2dt+1, i.e. they are in the Hilbert scheme of the rational normal curves of degree 2d. Thus, it turns out that they are natural generalizations of the double line in the plane considered as a degenerated conic.The simplest nontrivial example is the curve of degree 4 in ℙ⁴, defined by the ideal (xz−y ², xu−yv, yu−zv, u ², uv, v ²). The double rational curve allow the formulation of a Strong Castelnuovo Lemma in the sense of [7], for sets of points and double points. In the last section we mention some plethysm formulae for symmetric powers. Received: 1 September 2000 / Revised version: 15 January 2001     

Algebraic families of smooth hyperbolic surfaces of low degree in ℙ ℂ 3

We reprove (after a paper of Y.T. Siu appeared in 1987) a simple vanishing theorem for the Wronskian of Brody curves under a suitable assumption on the existence of global meromorphic connections. Next we give a slight improvement of a result due to Y.T, Siu and A.M. Nadel (Duke Math. J., 1989) on the algebraic degeneracy of entire holomorphic curves contained in certain hypersurfaces of ℙ _ℂ ⁿ . Especially, their result is generalized to a larger class of hypersurfaces. Our method produces algebraic families of smooth hyperbolic surfaces in ℙ _ℂ ³ for all degreesd≥14; this brings us somewhat nearer than previously known from the expected ranged≥5.     

Plane curves as projections of non singular space curves

We generalize and make rigorous a construction by Enriques which allows one to obtain a plane curve as the projection of a non singular curve spanning ℙ⁴ we show that every non singular curve in ℙ^r projecting onto a given plane curve can be obtained by the same construction. Finally we prove that every non singular plane curve of degree d is the projection of a (non singular) curve of degree 2d-1 spanning ℙ⁴, and that no lower degree is possible. Supported by the M. P. I. of the Italian Government     

On the divisor class group of double solids

For a double solid V→ℙ_3> branched over a surface B⊂ℙ₃(ℂ) with only ordinary nodes as singularities, we give a set of generators of the divisor class group in terms of contact surfaces of B with only superisolated singularities in the nodes of B. As an application we give a condition when H ^* (˜V , ℤ) has no 2-torsion. All possible cases are listed if B is a quartic. Furthermore we give a new lower bound for the dimension of the code of B. Received: 16 November 1998     

On first order congruences of lines of ℙ4 with a fundamental curve

We discuss projective families of lines of ℙ ⁿ , and in particular congruences of order one. After giving general results, we obtain a complete classification of the case of ℙ⁴ in which there is a fundamental curve. Received: 2 August 2000 / Revised version: 11 July 2001     

Invariant currents and dynamical Lelong numbers

Let ƒ be a polynomial automorphism of ℂ^k of degree λ, whose rational extension to ℙ^k maps the hyperplane at infinity to a single point. Given any positive closed current S on ℙ^k of bidegree (1,1), we show that the sequence λ⁻ⁿ(ƒⁿ)*S converges in the sense of currents on ℙ^k to a linear combination of the Green current T₊ of ƒ and the current of integration along the hyperplane at infinity. We give an interpretation of the coefficients in terms of generalized Lelong numbers with respect to an invariant dynamical current for ƒ⁻¹.     

Flex Curves and their Applications

In this paper, we define the notion of the flex curve F ⁽⁾(f; P) at a nonsingular point P of a plane curve Ca. We construct interesting plane curves using a cyclic covering transform, branched along F ⁽⁾(f; P). As an application, we show the moduli space of projective curves of degree 12 with 27 cusps has at least three irreducible components. Simultaneously, we give an example of Alexander-equivalent Zariski pair of irreducible curves.     

Postulation of subschemes of irreducible curves on a quadric surface

We study all the possible Hilbert functions of 0-dimensional subschemes of irreducible curves of a smooth quadric of ℙ³. We obtain characterizations in case of complete intersection, arithmetically Cohen-Macaulay and arithmetically Buchsbaum curves and other necessary conditions in the general cases.     

Index parity of closed geodesics and rigidity of Hopf fibrations

The diameter rigidity theorem of Gromoll and Grove [1987] states that a Riemannian manifold with sectional curvature ≥ 1 and diameter ≥ π/2 is either homeomorphic to a sphere, locally isometric to a rank one symmetric space, or it has the cohomology ring of the Cayley plane Caℙ. The reason that they were only able to recognize the cohomology ring of Caℙ is due to an exceptional case in another theorem [Gromoll and Grove, 1988]: A Riemannian submersion σ:?^m→B ^b with connected fibers that is defined on the Euclidean sphere ?^m is metrically congruent to a Hopf fibration unless possibly (m,b)=(15,8). We will rule out the exceptional cases in both theorems. Our argument relies on a rather unusal application of Morse theory. For that purpose we give a general criterion which allows to decide whether the Morse index of a closed geodesic is even or odd. Oblatum 7-II-2000 & 11-X-2000?Published online: 29 January 2001     

Laminar currents in ℙ<Superscript>2</Superscript>

 In this paper we study laminar currents in ℙ². Given a sequence of irreducible algebraic curves (C _n ) converging in the sense of currents to T, we find geometric conditions on the curves ensuring that the limit current T is laminar. This criterion is then applied to meromorphic dynamical systems in ℙ², and laminarity of the dynamical ``Green' current is obtained for a wide class of meromorphic self maps of ℙ², as well as for all bimeromorphic maps of projective surfaces. Received: 24 September 2001 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 32U40, 37Fxx, 32H50     

The picard group of the variety of plane cuspidal cubics of ?3

We give a compactification of the varietyU of non-degenerate plane cuspidal cubics of ℙ³. We construct this compactification by means of the projective bundleX of a suitable vector bundleE. We describe the intersection ring ofX and, as a consequence, we obtain the intersection numbers ofU that satisfy 10 conditions of the following kinds:ρ, that the plane determined by the cuspidal cubic go through a point;c, that the cusp be on a plane;q, that the cuspidal tangent intersect a line;μ, that the cuspidal cubic intersect a line. Moreover, we prove that the Picard group of the varietyU is a product of two infinite cyclic groups generated byρ andc−q.     

3-dimensional sundials

R. Hartshorne and A. Hirschowitz proved that a generic collection of lines on ℙ ⁿ , n≥3, has bipolynomial Hilbert function. We extend this result to a specialization of the collection of generic lines, by considering a union of lines and 3-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines).     

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     

The diffeomorphic types of the complements of arrangements in ??3 II

For any arrangement of hyperplanes in ℂℙ³, we introduce the soul of this arrangement. The soul, which is a pseudo-complex, is determined by the combinatorics of the arrangement of hyperplanes. In this paper, we give a sufficient combinatoric condition for two arrangements of hyperplanes to be diffeomorphic to each other. In particular we have found sufficient conditions on combinatorics for the arrangement of hyperplanes whose moduli space is connected. This generalizes our previous result on hyperplane point arrangements in ℂℙ³. This work was partially supported by NSA grant and NSF grant     

Steinberg triality groups acting on 2 ? (<Emphasis Type="Italic">v,k</Emphasis>, 1) designs

A 2 - (υ, k, 1) design D = (ℙ,ℙ, ℬ) is a system consisting of a finite set ℙ of υ points and a collection ℬ of ℙ-subsets of ℙ, called blocks, such that each 2-subset of ℙ is contained in precisely one block. Let G be an automorphism group of a 2-(υ, k, 1) design. Delandtsheer proved that if G is block-primitive and D is not a projective plane, then G is almost simple, that is, T ⩽ G ⩽ Aut(T), where T is a non-abelian simple group. In this paper, we prove that T is not isomorphic to ³ D ₄(q). This paper is part of a project to classify groups and designs where the group acts primitively on the blocks of the design.     

The genus of space curves

LetC be a curve contained in ℙ _k ³ (k of any characteristic), which is locally Cohen-Macaulay, not contained in a plane and of degreed. We prove thatp _a (C)≤≤((d−2)(d−3))/2. Moreover we show existence of curves with anyd, p _a satisfying this inequality and we characterize those curves for which equality holds.

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Sunto Si dimostra che seC⊃ℙ _k ³ (k di caratteristica qualunque) é una curva, non piana, localmente Cohen-Macaulay, di gradod, allorap _a (C)≤((d−2)(d−3))/2. Si mostra che questa limitazione è ottimale, e si classificano le curve di genere aritmetico massimale.

     

Rational cuspidal plane curves of type (d,d-4) with

In the present paper we classify rational cuspidal plane curves with maximal multiplicity deg C - 4 and at least three cusps and where (V,D) is the minimal (SNC) resolution of (ℙ²,C). Received: 28 August 1998     

Massless excitations of long strings in AdS4 × ?P3

We consider the AdS₄ × ℂℙ³ IIA superstring sigma model in the background of a “spinning string” classical solution with two charges. In the limit when one of the spins is infinite, there are massless excitations which govern the long-range worldsheet properties of the model. We obtain a sigma model of ℂℙ³ with fermions which describes the dynamics of these massless modes.     

On monomial k-Buchsbaum curves in ℙ3

Using the theory of affine semigroup rings we describe all strictly 2-Buchsbaum monomial curves in ℙ³.