191 research outputs found
Quatroids and Rational Plane Cubics
It is a classical result that there are (irreducible) rational cubic
curves through generic points in , but little is
known about the non-generic cases. The space of -point configurations is
partitioned into strata depending on combinatorial objects we call quatroids, a
higher-order version of representable matroids. We compute all
quatroids on eight distinct points in the plane, which produces a full
description of the stratification. For each stratum, we generate several
invariants, including the number of rational cubics through a generic
configuration. As a byproduct of our investigation, we obtain a collection of
results regarding the base loci of pencils of cubics and positive certificates
for non-rationality.Comment: 34 pages, 11 figures, 5 tables. Comments are welcome
On threefolds covered by lines
A classification theorem is given of projective threefolds that are covered
by a two-dimensional family of lines, but not by a higher dimensional family.Comment: LaTeX 2.09, 25 page
Counting rational points on projective varieties
We develop a global version of Heath-Brown\u27s p-adic determinant method to study the asymptotic behaviour of the number N(W; B) of rational points of height at most B on certain subvarieties W of Pn defined over Q. The most important application is a proof of the dimension growth conjecture of Heath-Brown and Serre for all integral projective varieties of degree d ≥ 2 over Q. For projective varieties of degree d ≥ 4, we prove a uniform version N(W; B) = Od,n,ε(BdimW+ε) of this conjecture. We also use our global determinant method to improve upon previous estimates for quasi-projective surfaces. If, for example, (Formula presented.) is the complement of the lines on a non-singular surface X ⊂\ua0P3 of degree d, then we show that (Formula presented.). For surfaces defined by forms (Formula presented.) with non-zero coefficients, then we use a new geometric result for Fermat surfaces to show that (Formula presented.) for B ≥ e
Reverse engineering of CAD models via clustering and approximate implicitization
In applications like computer aided design, geometric models are often
represented numerically as polynomial splines or NURBS, even when they
originate from primitive geometry. For purposes such as redesign and
isogeometric analysis, it is of interest to extract information about the
underlying geometry through reverse engineering. In this work we develop a
novel method to determine these primitive shapes by combining clustering
analysis with approximate implicitization. The proposed method is automatic and
can recover algebraic hypersurfaces of any degree in any dimension. In exact
arithmetic, the algorithm returns exact results. All the required parameters,
such as the implicit degree of the patches and the number of clusters of the
model, are inferred using numerical approaches in order to obtain an algorithm
that requires as little manual input as possible. The effectiveness, efficiency
and robustness of the method are shown both in a theoretical analysis and in
numerical examples implemented in Python
Monodromy invariants in symplectic topology
This text is a set of lecture notes for a series of four talks given at
I.P.A.M., Los Angeles, on March 18-20, 2003. The first lecture provides a quick
overview of symplectic topology and its main tools: symplectic manifolds,
almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants
and Floer homology. The second and third lectures focus on symplectic Lefschetz
pencils: existence (following Donaldson), monodromy, and applications to
symplectic topology, in particular the connection to Gromov-Witten invariants
of symplectic 4-manifolds (following Smith) and to Fukaya categories (following
Seidel). In the last lecture, we offer an alternative description of symplectic
4-manifolds by viewing them as branched covers of the complex projective plane;
the corresponding monodromy invariants and their potential applications are
discussed.Comment: 42 pages, notes of lectures given at IPAM, Los Angele
Investigating the BPS Spectrum of Non-Critical E_n Strings
We use the effective action of the non-critical strings to study its
BPS spectrum for . We show how to introduce mass parameters, or
Wilson lines, into the effective action, and then perform the appropriate
asymptotic expansions that yield the BPS spectrum. The result is the
character expansion of the spectrum, and is equivalent to performing the mirror
map on a Calabi-Yau with up to nine K\"ahler moduli. This enables a much more
detailed examination of the structure of the theory, and provides
extensive checks on the effective action description of the non-critical
string. We extract some universal ( independent) information concerning
the degeneracies of BPS excitations.Comment: 50 pages, harvmac (b
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