102 research outputs found
Multiplicities of Classical Varieties
The -multiplicity plays an important role in the intersection theory of
St\"uckrad-Vogel cycles, while recent developments confirm the connections
between the -multiplicity and equisingularity theory. In this paper
we establish, under some constraints, a relationship between the
-multiplicity of an ideal and the degree of its fiber cone. As a
consequence, we are able to compute the -multiplicity of all the ideals
defining rational normal scrolls. By using the standard monomial theory, we can
also compute the - and -multiplicity of ideals defining
determinantal varieties: The found quantities are integrals which, quite
surprisingly, are central in random matrix theory.Comment: 27 pages; to appear in Proc. London Math. So
Differentiating by prime numbers
We introduce -derivations and give a few basic ways in which they act like
derivatives by numbers
Mapping toric varieties into low dimensional spaces
A smooth d-dimensional projective variety X can always be embedded into 2d + 1-dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax our requirement and ask only that the map is injective, then any d-dimensional projective variety can be mapped injectively to 2d + 1-dimensional projective space. A natural question then arises: what is the minimal m such that a projective variety can be mapped injectively to m-dimensional projective space? In this paper we investigate this question for normal toric varieties, with our most complete results being for Segre-Veronese varieties
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