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.

     

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.     

On double coverings of hyperelliptic curves

We prove various properties of varieties of special linear systems on double coverings of hyperelliptic curves. We show and determine the irreducibility, generically reducedness and singular loci of the variety for bi-elliptic curves and double coverings of genus two curves. Similar results for double coverings of hyperelliptic curves of genus h≥3 are also presented.     

Self-linked curves and normal bundle

We give necessary conditions on the degree and the genus of a smooth, integral curve C⊂P³$C\subset {\mathbb{P}}^3$ to be self-linked (i.e. locus of simple contact of two surfaces). We also give similar results for set theoretically complete intersection curves with a structure of multiplicity three (i.e. locus of 2-contact of two surfaces).     

Imaginary automorphisms on real hyperelliptic curves

A real hyperelliptic curve X is said to be Gaussian if there is an automorphism such that , where [-1] denotes the hyperelliptic involution on X. Gaussian curves arise naturally in several contexts, for example when one studies real Jacobians. In the present paper, we study the properties of Gaussian curves and we describe their moduli spaces.     

On certain remarkable curves of genus five

The aim of this note is twofold. First to show the existence of genus five curves having exactly twenty four Weierstrass points, which constitute the set of fixed points of three distinct elliptic involutions on them. Second to characterize these curves, in fact we prove that all such curves are bielliptic double cover of Fermat's quartic.     

On the weight of numerical semigroups

We investigate the weights of a family of numerical semigroups by means of even gaps and the Weierstrass property for such a family.     

On a class of rational cuspidal plane curves

We obtain new examples and the complete list of the rational cuspidal plane curvesC with at least three cusps, one of which has multiplicitydegC-2. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.     

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.     

Stratification of the Hilbert scheme of space curves with a stable normal bundle

Ford≥3g and 1≤s≤[g/2], we study the strataN _{d, g(s)} of degreed genusg spaces curvesC whose normal bundleN _C is stable with stability degree (integer of Lange-Narasimhan) σ(N _C)=2s. We prove thatN _{d, g(s)} has an irreducible component of the right dimension whose general curve has a normal bundle with the right number of maximal subbundles. We consider also the semi-stable case (s=0), obtaining similar results. We prove our results by studying the normal bundles of reducible curves and their deformations. Both authors were partially supported by MIUR and GNSAGA of INdAM (Italy).     

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.     

Trigonal non-Gorenstein curves

We answer some questions on trigonal non-Gorenstein curves mainly equipped with a positive Maroni , such as the number of non-Gorenstein points, the kind of such singularities, possible canonical models, uniqueness and number of base points of such linear systems, and the amplitude of the Maroni invariant.     

Characterization of abelian varieties

We prove that any smooth complex projective variety X with plurigenera P ₁(X)=P ₂(X)=1 and irregularity q(X)=dim(X) is birational to an abelian variety. Oblatum 26-V-1999 & 13-VI-2000?Published online: 11 October 2000     

Trigonal reducible curves

Here we study the canonical model of a reducible trigonal Gorenstein curve X. We prove that the canonical model is arithmetically Cohen — Macaulay and lies in a minimal degree Hirzebruch surface, generalizing the classical theory of Maroni on smooth trigonal curves.