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Summer Graduate School

Commutative Algebra and its Interaction with Algebraic Geometry June 03, 2019 - June 14, 2019
Parent Program: --
Location: Center for Mathematics of University of Notre Dame
Organizers Craig Huneke (University of Virginia), Sonja Mapes (University of Notre Dame), Juan Migliore (University of Notre Dame), LEAD Claudia Polini (University of Notre Dame), Claudiu Raicu (University of Notre Dame)

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The figure represents a blow-up. The so called blow-up algebras or Rees rings are the algebraic realizations of such blow-ups.

Linkage is a method for classifying ideals in local rings. Residual intersections is a generalization of linkage to the case where the two `linked' ideals  need not have the same codimension. Residual intersections are ubiquitous: they play an important role in the study of blowups, branch and multiple point loci, secant varieties, and Gauss images; they appear naturally in intersection theory; and they have close connections with integral closures of ideals. 

Commutative algebraists have long used the Frobenius or p-th power map to study commutative rings containing a finite fi eld. The theory of tight closure and test ideals has widespread applications to the study of symbolic powers and to Briancon-Skoda type theorems for equi-characteristic rings.

Numerical conditions for the integral dependence of ideals and modules have a wealth of applications, not the least of which is in equisingularity theory. There is a long history of generalized criteria for integral dependence of ideals and modules based on variants of the Hilbert-Samuel and the Buchsbaum-Rim multiplicity that still require some remnants of finite length assumptions.

The Rees ring and the special fiber ring of an ideal arise in the process of blowing up a variety along a subvariety. Rees rings and special fiber rings also describe, respectively, the graphs and the images of rational maps between projective spaces. A difficult open problem in commutative algebra, algebraic geometry, elimination theory, and geometric modeling is to determine explicitly the equations defining graphs and images of rational maps.

The school will consist of the following four courses with exercise sessions plus a Macaulay2 workshop

  • Linkage and residual intersections
  • Characteristic p methods and applications
  • Blowup algebras
  • Multiplicity theory

Suggested prerequisites:

- An introduction to Commutative Algebra by Atiyah and MacDonald
- Commutative Algebra with a view towards algebraic geometry by David Eisenbud - Chapters 2-13 and 17-21


For eligibility and how to apply, see the Summer Graduate Schools homepage

Due to the small number of students supported by MSRI, only one student per institution will be funded by MSRI.

Keywords and Mathematics Subject Classification (MSC)
  • Linkage

  • residual intersections

  • multiplicities

  • Hilbert functions

  • Rees algebras

  • associated graded rings

  • tight closure

  • integral closure

  • symbolic powers

  • reductions

  • resolutions

  • Gorenstein rings

  • Cohen-Macaulay rings

  • canonical modules

  • local cohomology

  • Frobenious map

  • 7 F-signature

  • Hilbert multiplicity

  • Hilbert-Kunz multiplicity

  • strongly F-regular rings

  • etale fundamental group

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC