105 research outputs found
First Steps Towards Radical Parametrization of Algebraic Surfaces
We introduce the notion of radical parametrization of a surface, and we
provide algorithms to compute such type of parametrizations for families of
surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least
the degree minus 4) singularity, all irreducible surfaces of degree at most 5,
all irreducible singular surfaces of degree 6, and surfaces containing a pencil
of low-genus curves. In addition, we prove that radical parametrizations are
preserved under certain type of geometric constructions that include offset and
conchoids.Comment: 31 pages, 7 color figures. v2: added another case of genus
Generic Torelli theorem for Prym varieties of ramified coverings
In this paper we prove that the Prym map, from the space of double coverings
of a curve of genus g with r branch points to the moduli space of abelian
varieties, is generically injective if r>6 and g>1, r=6 and g>2, r=4 and g>4,
r=2 and g>5. We also show that a very generic Prym variety of dimension at
least 4 is not isogenous to a Jacobian.Comment: 24 pages, 2 figures. One reference added. To appear in Compositio
Mathematic
Seshadri constants of surfaces of degrees 6 and 8
We compute Seshadri constants \eps(X):= \eps(\O_X(1)) on surfaces
of degrees 6 and 8. Moreover, more generally, we prove that if is any
embedded surface of degree in \PP^r not containing lines,
then 1 < \eps(X) <2 if and only if the homogeneous ideal of is not
generated by only quadrics (in which case \eps(X)=3/2).Comment: 10 pages, 1 figure; to appear in IMRN with a more detailed
bibliograph
Rational curves on \bar{M}_g and K3 surfaces
Let be a smooth primitively polarized K3 surface of genus and
the fibration defined by a linear pencil in
. For general and , we work out the splitting type of the
locally free sheaf , where is the
modular morphism associated to . We show that this splitting type encodes
the fundamental geometrical information attached to Mukai's projection map
, where is
the stack parameterizing pairs with as above and a
stable curve. Moreover, we work out conditions on a fibration to induce a
modular morphism such that the normal sheaf is locally
free.Comment: published version, revisions in the exposition, minor mistakes
correcte
Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities
We characterize genus g canonical curves by the vanishing of combinatorial
products of g+1 determinants of Brill-Noether matrices. This also implies the
characterization of canonical curves in terms of (g-2)(g-3)/2 theta identities.
A remarkable mechanism, based on a basis of H^0(K_C) expressed in terms of
Szego kernels, reduces such identities to a simple rank condition for matrices
whose entries are logarithmic derivatives of theta functions. Such a basis,
together with the Fay trisecant identity, also leads to the solution of the
question of expressing the determinant of Brill-Noether matrices in terms of
theta functions, without using the problematic Klein-Fay section sigma.Comment: 35 pages. New results, presentation improved, clarifications added.
Accepted for publication in Math. An
The Canonical Model of a Singular Curve
We give refined statements and modern proofs of Rosenlicht's results about
the canonical model C' of an arbitrary complete integral curve C. Notably, we
prove that C and C' are birationally equivalent if and only if C is
nonhyperelliptic, and that, if C is nonhyperelliptic, then C' is equal to the
blowup of C with respect to the canonical sheaf \omega. We also prove some new
results: we determine just when C' is rational normal, arithmetically normal,
projectively normal, and linearly normal.Comment: 28 pages, no figures, IV Congresso Iberoamericano de Geometria
Complex
Brill–Noether general K3 surfaces with the maximal number of elliptic pencils of minimal degree
We explicitly construct Brill–Noether general K3 surfaces of genus 4, 6 and 8 having the maximal number of elliptic pencils of degrees 3, 4 and 5, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill–Noether general K3 surfaces of genus 4 and 6 without stable Lazarsfeld–Mukai bundles of minimal c2.publishedVersio
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