Osculating Degeneration of Curves

Abstract

The main objects of this paper are osculating spaces of order mto smooth algebraic curves, with the property of meeting the curve again. We prove that the only irreducible curves with an infinite number of this type of osculating spaces of order mare curves in P ^m+1whose degree nis greater than m + 1. This is a generalization of the result and proof of Kaji (Kaji, H. (1986). On the tangentially degenerate curves. J. London Math. Soc.33(2):430–440) that corresponds to the case m = 1. We also obtain an enumerative formula for the number of those osculating spaces to curves in P ^m+2. The case m = 1 of it is a classical formula proved with modern techniques by Le Barz (Le Barz, P. (1982). Formules multisécantes pour les courbes gauches quelconques. In: Enumerative Geometry and Classical Algebraic Geometry. Prog. in Mathematics 24, Birkhäuser, pp. 165–197).     

On the Bennequin Invariant and the Geometry of Wave Fronts

The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plücker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J ⁺ in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.     

A duality theorem for crossed products of hopf algebras

Abstract

We study the automorphism and collineation groups of plane curves, i.e., in ?², that are not necessarily smooth, and obtain bounds for these curves in terms of, their degree and the number of singularities. We also introduce the notion of bad curves and good curves, and show that all bad curves are rational.     

Recursions for characteristic numbers of genus one plane curves

Characteristic numbers of families of maps of nodal curves toP ² are defined as intersection of natural divisor classes. (This definition agrees with the usual definition for families of plane curves.) Simple recursions for characteristic numbers of genus one plane curves of all degrees are computed.     

ON THE COHOMOLOGY OF A SPACE CURVE CONTAINING A PLANE CURVE

In this paper we study the Hartshorne–Rao module of curves in P ³ of degree d and genus g, containing plane curves of degree d ? p, p ≥ 1. We prove an optimal upper bound for the Rao function of these curves and we show that the curves attaining the bound are obtained from an extremal curve by an elementary biliaison of height min(p, d ? p) ? 1 on a quadric surface.     

Rational curves of degree at most 9 on a general quintic threefold

ABSTRACT.

We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P ⁴. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (?1) ⊕ (?1), and in P ⁴ with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.     

An operator-theoretic approach to theorems of the Pick-Nevanlinna and Carathéodory types

In this paper we deal with singularities of the linear systems of plane curves passing through S, where S is a zerodimensional closed subscheme of degree n of P ²=P _k ² ,k an algebraically closed field of any characteristic. We determine the least degree of a nonsingular curve passing through S, when S is in uniform position. This paper was written while the author was member of C.N.R., Sez. 3 of G.N.S.A.G.A. and was supported by M.P.I. funds     

Curves on Generic Surfaces of High Degree Through a Complete Intersection in P 3

We consider general surfaces, S, of high degree containing a given complete intersection space curve, Y. We study integral curves in the subgroup of Pic(S) generated by Y and the plane section. We determine the cohomological invariants of these curves and classify the subcanonical ones. Then using these subcanonical curves we produce stable rank two vector bundles on P ³.     

The Most Symmetric Nonsingular Plane Curves of Degree n≤20, I*

If n is an odd prime less than 20, then the most symmetric nonsingular plane curves in P ² of degree n are projectively equivalent to the Fermat curve x ⁿ +y ⁿ +z ⁿ .     

Graph theoretic techniques in algebraic geometry II: construction of singular complex surfaces of the rational cohomology type of CP2

Methods of graph theory are used to obtain rational projective surfaces with only rational double points as singularities and with rational cohomology rings isomorphic to that of the complex projective plane. Uniqueness results for such cohomologyCP ²'s and for rational and integral homologyCP ²'s are given in terms of the typesA _k,D _k, orE _k of singularities allowed by the construction. Supported in part by National Science Foundation grant no. MCS 77-03540.     

Limit laws for embedded trees: Applications to the integrated superBrownian excursion

We study three families of labeled plane trees. In all these trees, the root is labeled 0 and the labels of two adjacent nodes differ by 0,1, or ?1. One part of the paper is devoted to enumerative results. For each family, and for all j ∈ ?, we obtain closed form expressions for the following three generating functions: the generating function of trees having no label larger than j; the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labeled j; and finally the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labeled at least, j. Strangely enough, all these series turn out to be algebraic, but we have no combinatorial intuition for this algebraicity. The other part of the paper is devoted to deriving limit laws from these enumerative results. In each of our families of trees, we endow the trees of size n with the uniform distribution and study the following random variables: M_n, the largest label occurring in a (random) tree; X_n(j), the number of nodes labeled j; and X(j), the number of nodes labeled j or more. We obtain limit laws for scaled versions of these random variables. Finally, we translate the above limit results into statements dealing with the integrated superBrownian excursion. In particular, we describe the law of the supremum of its support (thus recovering some earlier results obtained by Delmas) and the law of its distribution function at a given point. We also conjecture the law of its density (at a given point). © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Equianalytic and equisingular families of curves on surfaces

Summary We consider flat families of reduced curves on a smooth surfaceS such that each memberC has the same number of singularities and each singularity has a fixed singularity type (up to analytic resp. topological equivalence). We show that these families are represented by a schemeH and give sufficient conditions for the smoothness ofH (atC). Our results improve previously known criteria for families with fixed analytic singularity type and seem to be quite sharp for curves in ℙ² of small degree. Moreover, for families with fixed topological type this paper seems to be the first in which arbitrary singularities are treated. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

Nagata’s conjecture for a square or nearly-square number of points

Abstract In this paper we prove that a linear system of plane curves of degree d with n points of multiplicity m always has the expected dimension when n is a perfect square. We also prove some results on the dimension of the linear system when n is close to a perfect square. Keywords: Interpolation, Nagata, Linear systems, Plane curves Mathematics Subject Classification (2000): 14J26, 14N05     

Remarks on Type III Unprojection

In this note we deal with rational curves in ? ³ which are images of a line by means of a finite sequence of cubo-cubic Cremona transformations. We prove that these curves can always be obtained applying to the line a sequence of such transformations increasing at each step the degree of the curve. As a corollary we get a result about curves that can give speciality for linear systems of ? ³.     

Cutting Circles into Pseudo-Segments and Improved Bounds for Incidences% and Complexity of Many Faces

Abstract. We show that n arbitrary circles in the plane can be cut into O(n ^3/2+ɛ ) arcs, for any ɛ>0 , such that any pair of arcs intersects at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree.     

Cutting Circles into Pseudo-Segments and Improved Bounds for Incidences% and Complexity of Many Faces

   Abstract. We show that n arbitrary circles in the plane can be cut into O(n ^3/2+ɛ ) arcs, for any ɛ>0 , such that any pair of arcs intersects at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree.     

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