52,803 research outputs found
Universal families of arcs and curves on surfaces
The main goal of this paper is to investigate the minimal size of families of
curves on surfaces with the following property: a family of simple closed
curves on a surface realizes all types of pants decompositions if for
any pants decomposition of the surface, there exists a homeomorphism sending it
to a subset of the curves in . The study of such universal families of
curves is motivated by questions on graph embeddings, joint crossing numbers
and finding an elusive center of moduli space. In the case of surfaces without
punctures, we provide an exponential upper bound and a superlinear lower bound
on the minimal size of a family of curves that realizes all types of pants
decompositions. We also provide upper and lower bounds in the case of surfaces
with punctures which we can consider labelled or unlabelled, and investigate a
similar concept of universality for triangulations of polygons, where we
provide bounds which are tight up to logarithmic factors.Comment: v2: Fixed a mistake in one of the lower bound
Minimal sections of conic bundles
Let the threefold X be a general smooth conic bundle over the projective
plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this
paper we prove the existence of two natural families C(+) and C(-) of curves on
X, such that the Abel-Jacobi map F sends one of these families onto a copy of
the theta divisor (Theta), and the other -- onto the jacobian J(X). The general
curve C of any of these two families is a section of the conic bundle
projection, and our approach relates such C to a maximal subbundle of a rank 2
vector bundle E(C) on C, or -- to a minimal section of the ruled surface
P(E(C)). The families C(+) and C(-) correspond to the two possible types of
versal deformations of ruled surfaces over curves of fixed genus g(C). As an
application, we find parameterizations of J(X) and (Theta) for certain classes
of Fano threefolds, and study the sets Sing(Theta) of the singularities of
(Theta).Comment: Duke preprint, 29 pages. LaTex 2.0
Surfaces of general type with and bicanonical map of degree 2
We classify the minimal algebraic surfaces of general type with and bicanonical map of degree 2. It will turn out that they are
isogenous to a product of curves, so that if is such a surface then there
exist two smooth curves and a finite group acting freely on such that . We describe the and that
occur. In particular the curve is a hyperelliptic-bielliptic curve of genus
3, and the bicanonical map of is composed with the involution
induced on by , where is the hyperelliptic involution of . In this way we obtain
three families of surfaces with which yield the first known
examples of surfaces with these invariants. We compute their dimension, and we
show that they are three smooth and irreducible components of the moduli space
of surfaces with . For each of these families, an
alternative description as a double cover of the plane is also given, and the
index of the paracanonical system is computed.Comment: 36 pages. To appear in Transactions of the American Mathematical
Societ
Surfaces of Degree 10 in the Projective Fourspace via Linear Systems and Linkage
The paper discusses the classification of surfaces of degree 10 and sectional
genus 9 and 10. The surfaces of degree at most 9 are described through
classical work dating from the last century up to recent years, while surfaces
of degree 10 and other sectional genera are studied elsewhere.
We use relations between multisecants, linear systems, syzygies and linkage
to describe the geometry of each surface. We want in fact to stress the
importance of multisecants and syzygies for the study of these surfaces.
Adjunction, which provided efficient arguments for the classification of
surfaces of smaller degrees, here appears to be less effective and will play
almost no role in the proofs.
We show that there are 8 different families of smooth surfaces of degree 10
and sectional genus 9 and 10. The families are determined by numerical data
such as the sectional genus, the Euler characteristic, the number of 6-secants
to the surface and the number of 5-secants to the surface which meet a general
plane. For each type we describe the linear system giving the embedding in P^4,
the resolution of the ideal, the geometry of the surface in terms of curves on
the surface and hypersurfaces containing the surface, and the liaison class; in
particular we find minimal elements in the even liaison class. Each type
corresponds to an irreducible, unirational component of the Hilbert scheme, and
the dimension is computed.Comment: 52 pages, plain Te
Associative Submanifolds of the 7-Sphere
Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal
submanifolds which are the links of calibrated 4-dimensional cones in R^8
called Cayley cones. Examples of associative 3-folds are thus given by the
links of complex and special Lagrangian cones in C^4, as well as Lagrangian
submanifolds of the nearly K\"ahler 6-sphere.
By classifying the associative group orbits, we exhibit the first known
explicit example of an associative 3-fold in S^7 which does not arise from
other geometries. We then study associative 3-folds satisfying the curvature
constraint known as Chen's equality, which is equivalent to a natural pointwise
condition on the second fundamental form, and describe them using a new family
of pseudoholomorphic curves in the Grassmannian of 2-planes in R^8 and
isotropic minimal surfaces in S^6. We also prove that associative 3-folds which
are ruled by geodesic circles, like minimal surfaces in space forms, admit
families of local isometric deformations. Finally, we construct associative
3-folds satisfying Chen's equality which have an S^1-family of global isometric
deformations using harmonic 2-spheres in S^6.Comment: 42 pages, v2: minor corrections, streamlined and improved exposition,
published version; Proceedings of the London Mathematical Society, Advance
Access published 17 June 201
Product-Quotient Surfaces: new invariants and algorithms
In this article we suggest a new approach to the systematic, computer-aided
construction and to the classification of product-quotient surfaces,
introducing a new invariant, the integer gamma, which depends only on the
singularities of the quotient model X=(C_1 x C_2)/G. It turns out that gamma is
related to the codimension of the subspace of H^{1,1} generated by algebraic
curves coming from the construction (i.e., the classes of the two fibers and
the Hirzebruch-Jung strings arising from the minimal resolution of
singularities of X).
Profiting from this new insight we developped and implemented an algorithm
which constructs all regular product-quotient surfaces with given values of
gamma and geometric genus in the computer algebra program MAGMA. Being far
better than the previous algorithms, we are able to construct a substantial
number of new regular product-quotient surfaces of geometric genus zero. We
prove that only two of these are of general type, raising the number of known
families of product-quotient surfaces of general type with genus zero to 75.
This gives evidence to the conjecture that there is an effective bound of the
form gamma < Gamma(p_g,q).Comment: 33 pages, 3 figure
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