Joins and Intersections

Abstract

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Lectures on Algebraic Statistics

“…See [48,105] for two book length monographs highlighting the connections of joins and secant varieties to classical algebraic geometry. Most of the algebraic geometry literature on secant varieties focuses on their dimensions, namely, to what extent this dimension can differ from the expected dimension.…”

Multiplicities and Rees valuations

“…Since the x i are analytically independent, M := (m, t −1 )R I is a height one prime ideal and R I /M is a polynomial ring in d variables over k. Thus, M is the only prime in R I of dimension d containing t −1 and m. Set T := (R I ) M . By the associativity formula for j-multiplicity (see [6], Proposition 6.1.3), it follows that…”

“…Thus, by definition, the j-multiplicity of I is given by j(I) := j(G I (R)) = j(R I /t −1 R I ), where j(G I (R)) is the normalized coefficient of the degree d − 1 term in the Hilbert polynomial of the G I (R)-module Γ m (G I (R))) (see [1] or [6], Section 6.1). Since this polynomial has degree d − 1 if and only if Γ m (G I (R))) has dimension d if and only if the analytic spread of I equals d, our attention will be focused primarily on ideals with this latter property.…”

Self-Intersections of Surfaces and Whitney Stratifications

“…If one wants to use the Stückrad-Vogel intersection cycle of a self-intersection of a surface X in P 3 for the construction of a Whitney stratification of X, then it seems to be natural to consider v(X, X, X) or v(v(X, X), X) rather than v(X, X), since the latter cycle cannot have a 0-dimensional part (see [8,Remark 2…”

“…Thus g(x) = m x (X) + · · · = 1, which implies m x (X) = 1. Conversely, the regularity of O X,x implies the regularity of A = OĴ ,x ∆ (see, for example, [8,Corollary 1.3.15]) and of A/I ∼ = OX ,x , where we used our notation introduced before Proposition 3.6. Hence I is generated by regular parameters, G I (A) is a polynomial ring over the regular local ring A/I, and g(x) = e(G I (A)) = 1.…”