98 research outputs found
Severi varieties and branch curves of abelian surfaces of type (1,3)
Let (A,L) be a principally polarized abelian surface of type (1,3). The
linear system |L| defines a 6:1 covering of A onto P2, branched along a curve B
of degree 18 in P2. The main result of the paper is that for general (A,L) the
curve B irreducible, admits 72 cusps, 36 nodes or tacnodes, each tacnode
counting as two nodes, 72 flexes and 36 bitangents. The main idea of the proof
is to use the fact that for a general (A,L) of type (1,3) the closure of the
Severi variety V in |L| is dual to the curve B in the sense of projective
geometry. We investigate V and B via degeneration to a special abelian surface.Comment: 17 page
Excess dimension for secant loci in symmetric products of curves
We extend a result of W. Fulton, J. Harris and R. Lazarsfeld to secant loci
in symmetric products of curves. We compare three secant loci and prove the the
dimensions of bigger loci can not be excessively larger than the dimension of
smaller loci.Comment: final version, to appear in Collectanea Mat
The locus of points of the Hilbert scheme with bounded regularity
In this paper we consider the Hilbert scheme parameterizing
subschemes of with Hilbert polynomial , and we investigate its
locus containing points corresponding to schemes with regularity lower than or
equal to a fixed integer . This locus is an open subscheme of
and, for every , we describe it as a locally closed
subscheme of the Grasmannian given by a set of equations of
degree and linear inequalities in the coordinates
of the Pl\"ucker embedding.Comment: v2: new proofs relying on the functorial definition of the Hilbert
scheme. v3: Sections reorganized, new self-contained proof of the
representability of the Hilbert functor with bounded regularity (Section 6
On the hypersurface of Luroth quartics
The hypersurface of Luroth quartic curves inside the projective space of plane quartics has degree 54. We give a proof of this fact along the lines outlined in a paper by Morley, published in 1919. Another proof has been given by Le Potier and Tikhomirov in 2001, in the setting of moduli spaces of vector bundles on the projective plane. Morley's proof uses the description of plane quartics as branch curves of Geiser involutions and gives new geometrical interpretations of the 36 planes associated to the Cremona hexahedral representations of a nonsingular cubic surface
The curve of lines on a prime Fano threefold of genus 8
We show that a general prime Fano threefold X of genus 8 can be reconstructed
from the pair , where is its Fano curve of lines and
is the theta-characteristic which gives a natural embedding
\Gamma \subset \matbb{P}^5.Comment: 24 pages, misprints corrected, to appear in International Journal of
Mathematic
PALEORADIOLOGICAL STUDY ON TWO INFANTS DATED TO THE 17th AND 18th CENTURIES
During an excavation campaign in the Church of the Conversion of Saint Paul in Roccapelago (North Italy), a hidden crypt was discovered, which yielded the remains of more than 400 individuals. The crypt was used as a cemetery by the inhabitants of the village of Roccapelago between the 16th and 18th centuries. Along the north side of the crypt, an area apparently separated from the rest of the burials was found, bordered by stones, where several burials of newborns and infants were concentrated. From here, five fabric rolls containing bones were recovered, and it was decided not to carry out destructive analyses, allocating the two best examples to a thorough radiological investigation to try to define the type of burial and the complete biological profile of the infant. The two rolls, subjects of this study, can be dated archaeologically between the 17th and 18th centuries. CT analysis shows a varied group of bones with a fairly good state of conservation. The paleoradiological study carried out had the primary objective of avoiding the destruction of the two rolls, ensuring their conservation; but at the same time, providing essential data to understand their nature, defining the biological profile and the type of deposition
Chen-Ruan cohomology of ADE singularities
We study Ruan's \textit{cohomological crepant resolution conjecture} for
orbifolds with transversal ADE singularities. In the -case we compute both
the Chen-Ruan cohomology ring and the quantum corrected
cohomology ring . The former is achieved in general, the
later up to some additional, technical assumptions. We construct an explicit
isomorphism between and in the -case,
verifying Ruan's conjecture. In the -case, the family
is not defined for . This implies that
the conjecture should be slightly modified. We propose a new conjecture in the
-case which we prove in the -case by constructing an explicit
isomorphism.Comment: This is a short version of my Ph.D. Thesis math.AG/0510528. Version
2: chapters 2,3,4 and 5 has been rewritten using the language of groupoids; a
link with the classical McKay correpondence is given. International Journal
of Mathematics (to appear
The Noether\u2013Lefschetz locus of surfaces in toric threefolds
The Noether-Lefschetz theorem asserts that any curve in a very general surface (Formula presented.) in (Formula presented.) of degree (Formula presented.) is a restriction of a surface in the ambient space, that is, the Picard number of (Formula presented.) is (Formula presented.). We proved previously that under some conditions, which replace the condition (Formula presented.), a very general surface in a simplicial toric threefold (Formula presented.) (with orbifold singularities) has the same Picard number as (Formula presented.). Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in (Formula presented.) in a linear system of a Cartier ample divisor with respect to a (Formula presented.)-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether-Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense
Deformation of canonical morphisms and the moduli of surfaces of general type
In this article we study the deformation of finite maps and show how to use
this deformation theory to construct varieties with given invariants in a
projective space. Among other things, we prove a criterion that determines when
a finite map can be deformed to a one--to--one map. We use this criterion to
construct new simple canonical surfaces with different and . Our
general results enable us to describe some new components of the moduli of
surfaces of general type. We also find infinitely many moduli spaces having one component whose general point corresponds to a
canonically embedded surface and another component whose general point
corresponds to a surface whose canonical map is a degree 2 morphism.Comment: 32 pages. Final version with some simplifications and clarifications
in the exposition. To appear in Invent. Math. (the final publication is
available at springerlink.com
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