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DTSTART;TZID=America/New_York:20260409T160000
DTEND;TZID=America/New_York:20260409T170000
DTSTAMP:20260409T015638
CREATED:20260310T200849Z
LAST-MODIFIED:20260310T200849Z
UID:10003081-1775750400-1775754000@www.math.harvard.edu
SUMMARY:Multiplicities of graded families of ideals on Noetherian local rings
DESCRIPTION:Let $R$ be an arbitrary $d$-dimensional Noetherian local ring with maximal ideal $m_R$. In this talk\, we give a generalization of the multiplicity $e(I)$ of an $m_R$-primary ideal $I$ of $R$ to a multiplicity $e(\mathcal I)$ of a graded family of $m_R$-primary ideals $\mathcal I$ in $R$. This multiplicity gives the classical multiplicity $e(I)$ if $\mathcal I=\{I^n\}$ is the $I$-adic filtration\, and agrees with the volume\, $\lim_{n\rightarrow \infty}d!\frac{\ell(R/I_n) }{n^d}$ for $R$ such that $\dim N(\hat R)>d$\, the required condition for the volume of graded families of $m_R$-primary ideals to exist as a limit. We will show that many of the classical theorems for the multiplicity of an ideal generalize to this multiplicity\, including mixed multiplicities\, the Rees theorem and the Minkowski inequality and equality. We give proofs which are independent of the theory of volumes and Okounkov bodies for all of our results\, with the one exception being the proof of the Minkowski equality. We do this by interpreting the multiplicity of graded families of $m_R$-primary ideals as an intersection product on the family of $R$-schemes which are obtained by blowing up $m_R$-primary ideals in $R$.
URL:https://www.math.harvard.edu/event/multiplicities-of-graded-families-of-ideals-on-noetherian-local-rings/
LOCATION:CMSA\, 20 Garden St\, G10\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:CMSA Algebra Seminar
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DTSTART;TZID=America/New_York:20260409T160000
DTEND;TZID=America/New_York:20260409T180000
DTSTAMP:20260409T015638
CREATED:20260126T151247Z
LAST-MODIFIED:20260406T135314Z
UID:10003011-1775750400-1775757600@www.math.harvard.edu
SUMMARY:Rational Maps on Elliptic Surfaces
DESCRIPTION:We will discuss rational self-maps on surfaces which respect (but do not necessarily fix) an elliptic fibration on the surface i.e. fibres are mapped to fibres. Such elliptic surfaces arguably form the largest swathe of the classification of surfaces\, and these maps\, elliptic skew products\, likewise form a significant class of rational maps. However\, little has been said about their dynamics. First I will show that these maps fall into two clean categories. Intriguing algebraic and arithmetic questions follow\, which I will share and answer to a greater extent. Using a mix of algebraic\, geometric\, arithmetic\, and dynamical arguments we will work toward a descriptive classification. Joint work in progress with Giacomo Mezzedimi (Bonn).\n\nGo to http://people.math.harvard.edu/~demarco/AlgebraicDynamics/ for more information
URL:https://www.math.harvard.edu/event/algebraic-dynamics-seminar-richard-birkett-brown-university/
LOCATION:Science Center 232
CATEGORIES:ALGEBRAIC DYNAMICS
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