182 research outputs found
Graph hypersurfaces and a dichotomy in the Grothendieck ring
The subring of the Grothendieck ring of varieties generated by the graph
hypersurfaces of quantum field theory maps to the monoid ring of stable
birational equivalence classes of varieties. We show that the image of this map
is the copy of Z generated by the class of a point. Thus, the span of the graph
hypersurfaces in the Grothendieck ring is nearly killed by setting the
Lefschetz motive L to zero, while it is known that graph hypersurfaces generate
the Grothendieck ring over a localization of Z[L] in which L becomes
invertible. In particular, this shows that the graph hypersurfaces do not
generate the Grothendieck ring prior to localization. The same result yields
some information on the mixed Hodge structures of graph hypersurfaces, in the
form of a constraint on the terms in their Deligne-Hodge polynomials.Comment: 8 pages, LaTe
Algorithms to Compute Characteristic Classes
In this thesis we develop several new algorithms to compute characteristics classes in a variety of settings. In addition to algorithms for the computation of the Euler characteristic, a classical topological invariant, we also give algorithms to compute the Segre class and Chern-Schwartz-MacPherson (CSM) class. These invariants can in turn be used to compute other common invariants such as the Chern-Fulton class (or the Chern class in smooth cases).
We begin with subschemes of a projective space over an algebraically closed field of characteristic zero. In this setting we give effective algorithms to compute the CSM class, Segre class and the Euler characteristic. The algorithms can be implemented using either symbolic or numerical methods. The algorithms are based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Running time bounds are given for these algorithms and the algorithms are found to perform favourably compared to other applicable algorithms. Relations between our algorithms and other existing algorithms are explored. In the special case of a complete intersection subcheme we develop a second algorithm to compute CSM classes and Euler characteristics in a more direct and efficient manner.
Each of these algorithms are generalized to subschemes of a product of projective spaces. Running time bounds for the generalized algorithms to compute the CSM class, Segre class and the Euler characteristic are given. Our Segre class algorithm is tested in comparison to another applicable algorithm and is found to perform favourably. To the best of our knowledge there are no other algorithms in the literature which compute the CSM class and Euler characteristic in the multi-projective setting.
For complete simplical toric varieties defined by a fan we give a strictly combinatorial algorithm to compute the CSM class and Euler characteristic and a second combinatorial algorithm with reduced running time to compute only the Euler characteristic.
We also prove several Bezout type bounds in multi-projective space. An application of these bounds to obtain a sharper degree bound on a certain system with a natural bi-projective structure is demonstrated
Feynman integral relations from parametric annihilators
We study shift relations between Feynman integrals via the Mellin transform
through parametric annihilation operators. These contain the momentum space IBP
relations, which are well-known in the physics literature. Applying a result of
Loeser and Sabbah, we conclude that the number of master integrals is computed
by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate
techniques to compute this Euler characteristic in various examples and compare
it with numbers of master integrals obtained in previous works.Comment: v2: new section 3.1 added, several misprints corrected and additional
remark
Shadows of blow-up algebras
We study different notions of blow-up of a scheme X along a subscheme Y,
depending on the datum of an embedding of X into an ambient scheme. The two
extremes in this theory are the ordinary blow-up, corresponding to the
identity, and the `quasi-symmetric blow-up', corresponding to an embedding into
a nonsingular variety M. We prove that this latter blow-up is intrinsic of Y
and X, and is universal with respect to the requirement of being embedded as a
subscheme of the ordinary blow-up of some ambient space along Y.
We consider these notions in the context of the theory of characteristic
classes of singular varieties. We prove that if X is a hypersurface in a
nonsingular variety and Y is its `singularity subscheme', these two extremes
embody respectively the conormal and characteristic cycles of X. Consequently,
the first carries the essential information computing Chern-Mather classes, and
the second is likewise a carrier for Chern-Schwartz-MacPherson classes. In our
approach, these classes are obtained from Segre class-like invariants, in
precisely the same way as other intrinsic characteristic classes such as those
proposed by William Fulton and by W. Fulton and Kent Johnson.
We also identify a condition on the singularities of a hypersurface under
which the quasi-symmetric blow-up is simply the linear fiber space associated
with a coherent sheaf.Comment: 23 pages. Substantial revision of the January 2002 versio
Elliptic double affine Hecke algebras
We give a construction of an affine Hecke algebra associated to any Coxeter
group acting on an abelian variety by reflections; in the case of an affine
Weyl group, the result is an elliptic analogue of the usual double affine Hecke
algebra. As an application, we use a variant of the version of
the construction to construct a flat noncommutative deformation of the th
symmetric power of any rational surface with a smooth anticanonical curve, and
give a further construction which conjecturally is a corresponding deformation
of the Hilbert scheme of points.Comment: 134 pages. v2: Added results on centers and generic Morita
equivalence, plus a description of a conjectural construction of deformed
Hilbert scheme
FDOA-based passive source localization: a geometric perspective
2018 Fall.Includes bibliographical references.We consider the problem of passively locating the source of a radio-frequency signal using observations by several sensors. Received signals can be compared to obtain time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements. The geometric relationship satisfied by these measurements allow us to make inferences about the emitter's location. In this research, we choose to focus on the FDOA-based source localization problem. This problem has been less widely studied and is more difficult than solving for an emitter's location using TDOA measurements. When the FDOA-based source localization problem is formulated as a system of polynomials, the source's position is contained in the corresponding algebraic variety. This provides motivation for the use of methods from algebraic geometry, specifically numerical algebraic geometry (NAG), to solve for the emitter's location and gain insight into this system's interesting structure
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