12,626 research outputs found
Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian
We prove the existence of various families of irreducible homaloidal
hypersurfaces in projective space , for all . Some of
these are families of homaloidal hypersurfaces whose degrees are arbitrarily
large as compared to the dimension of the ambient projective space. The
existence of such a family solves a question that has naturally arisen from the
consideration of the classes of homaloidal hypersurfaces known so far. The
result relies on a fine analysis of dual hypersurfaces to certain scroll
surfaces. We also introduce an infinite family of determinantal homaloidal
hypersurfaces based on a certain degeneration of a generic Hankel matrix. These
examples fit non--classical versions of de Jonqui\`eres transformations. As a
natural counterpoint, we broaden up aspects of the theory of Gordan--Noether
hypersurfaces with vanishing Hessian determinant, bringing over some more
precision to the present knowledge.Comment: 56 pages. Some material added in section 1; minor changes. Final
version to appear in Advances in Mathematic
Polar degrees and closest points in codimension two
Suppose that is a toric variety of codimension
two defined by an integer matrix , and let be a Gale
dual of . In this paper we compute the Euclidean distance degree and polar
degrees of (along with other associated invariants) combinatorially
working from the matrix . Our approach allows for the consideration of
examples that would be impractical using algebraic or geometric methods. It
also yields considerably simpler computational formulas for these invariants,
allowing much larger examples to be computed much more quickly than the
analogous combinatorial methods using the matrix in the codimension two
case.Comment: 25 pages, 1 figur
The bottleneck degree of algebraic varieties
A bottleneck of a smooth algebraic variety is a pair
of distinct points such that the Euclidean normal spaces at
and contain the line spanned by and . The narrowness of bottlenecks
is a fundamental complexity measure in the algebraic geometry of data. In this
paper we study the number of bottlenecks of affine and projective varieties,
which we call the bottleneck degree. The bottleneck degree is a measure of the
complexity of computing all bottlenecks of an algebraic variety, using for
example numerical homotopy methods. We show that the bottleneck degree is a
function of classical invariants such as Chern classes and polar classes. We
give the formula explicitly in low dimension and provide an algorithm to
compute it in the general case.Comment: Major revision. New introduction. Added some new illustrative lemmas
and figures. Added pseudocode for the algorithm to compute bottleneck degree.
Fixed some typo
How Important is the Currency Denomination of Exports in Open-Economy Models?
We show that standard alternative assumptions about the currency in which firms price export goods are virtually inconsequential for the properties of aggregate variables, other than the terms of trade, in a quantitative open-economy model. This result is in contrast to a large literature that emphasizes the importance of the currency denomination of exports for the properties of open-economy models.local currency pricing; producer currency pricing; international relative prices; exchange rates; nontraded goods; distribution services
How important is the currency denomination of exports in open-economy models?
The authors show that standard alternative assumptions about the currency in which firms price export goods are virtually inconsequential for the properties of aggregate variables, other than the terms of trade, in a quantitative open-economy model. This result is in contrast to a large literature that emphasizes the importance of the currency denomination of exports for the properties of open-economy models.Exports ; Pricing
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