7,686 research outputs found
Monodromy of Projective Curves
The uniform position principle states that, given an irreducible
nondegenerate curve C in the projective r-space , a general (r-2)-plane L
is uniform, that is, projection from L induces a rational map from C to
whose monodromy group is the full symmetric group. In this paper we show the
locus of non-uniform (r-2)-planes has codimension at least two in the
Grassmannian for a curve C with arbitrary singularities. This result is optimal
in . For a smooth curve C in that is not a rational curve of degree
three, four or six, we show any irreducible surface of non-uniform lines is a
Schubert cycle of lines through a point , such that projection from is
not a birational map of onto its image.Comment: corrected typo in first paragraph of introduction, 23 pages, AMSLaTe
Vector bundles and theta functions on curves of genus 2 and 3
Let C be a curve of genus g, and let SU(r) be the moduli space of vector
bundles of rank r on C, with trivial determinant. A general E in SU(r) defines
a theta divisor in the linear system |r Theta|, where Theta is the canonical
theta divisor in Pic^{g-1}(C). This defines a rational map SU(r) - - > |r
Theta|, which is the map associated to the determinant bundle on SU(r) (the
positive generator of Pic(SU(r)). In this paper we prove that in genus 2 this
map is generically finite and dominant. The same method, together with some
classical work of Morin, shows that in rank 3 and genus 3 the theta map is a
finite morphism -- in other words, every E in SU(3) admits a theta divisor.Comment: 12 page
Addendum to `Fake Projective Planes'
The addendum updates the results presented in the paper `Fake Projective
Plane, Invent Math 168, 321-370 (2007)' and makes some additions and
corrections. The fake projective planes are classified into twenty six classes.
Together with a recent work of Donald Cartwright and Tim Steger, there is now a
complete list of fake projective planes. There are precisely one hundred fake
projective planes as complex surfaces classified up to biholomorphism.Comment: A more refined classification is given in the new versio
Characteristic varieties and Betti numbers of free abelian covers
The regular \Z^r-covers of a finite cell complex X are parameterized by the
Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording
when the Betti numbers b_1,..., b_i of the corresponding covers are finite
carves out certain subsets \Omega^i_r(X) of the Grassmannian.
We present here a method, essentially going back to Dwyer and Fried, for
computing these sets in terms of the jump loci for homology with coefficients
in rank 1 local systems on X. Using the exponential tangent cones to these jump
loci, we show that each \Omega-invariant is contained in the complement of a
union of Schubert varieties associated to an arrangement of linear subspaces in
H^1(X,\Q).
The theory can be made very explicit in the case when the characteristic
varieties of X are unions of translated tori. But even in this setting, the
\Omega-invariants are not necessarily open, not even when X is a smooth complex
projective variety. As an application, we discuss the geometric finiteness
properties of some classes of groups.Comment: 40 pages, 2 figures; accepted for publication in International
Mathematics Research Notice
Light dual multinets of order six in the projective plane
The aim of this paper is twofold: First we classify all abstract light dual
multinets of order which have a unique line of length at least two. Then we
classify the weak projective embeddings of these objects in projective planes
over fields of characteristic zero. For the latter we present a computational
algebraic method for the study of weak projective embeddings of finite
point-line incidence structures
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