# Commutative Algebra and its Interaction with Algebraic Geometry (07w5505)

Arriving in Banff, Alberta Sunday, June 10 and departing Friday June 15, 2007

## Organizers

Anthony Geramita (Queens University)

Paul Roberts (University of Utah)

Bernd Ulrich (Purdue University)

## Objectives

1. Problems in positive and mixed characteristic.

For rings of prime characteristic, the Frobenius map, which sends an element of the ring to its pth power, where p is the characteristic of the ring, is an extremely powerful tool. Many conjectures on the

homological theory of Noetherian rings have been proven using this technique, and more recently these ideas have been extended in the theory of tight closure. For rings of mixed characteristic, an

important line of research has been to prove these conjectures and to extend these ideas to that case.

Positive characteristic, tight closure, and Hilbert-Kunz multiplicities.

The theory of tight closure was introduced by Hochster and Huneke a number of years ago and is still a very active area. Connections have been proven to exist between the concepts which arise naturally

in this theory and classical properties of singularities by Karen Smith, Watanabe, and others, and there is now a considerable amount of interest in relations with deep properties of local cohomology. Among the current topics of research in this area are test ideals, preservation under base change, and relations with local cohomology, including work by Smith, Lyubeznik, and Singh. A recent development has been the relation of the action of the Frobenius map on local cohomology to connectedness theorems. Another topic is the connection with multiplier ideals, an issue we discuss in more detail in the second section. Recently there has been a lot of progress on multiplier ideals, arc spaces and valuations in work of Ein, Mustata, and others; the connections with valuation theory are a classical topic that runs

through all areas of this subject.

The other topic of this section is Hilbert-Kunz multiplicities. These invariants are an analogue of traditional multiplicities in Algebraic Geometry, which are discussed in the second section of this proposal, and they record the action of the Frobenius homomorphism on a ring in positive characteristic. They are related to local Chern classes and other arithmetic invariants. Recent results by Brenner and Trivedi use results on vector bundles to compute these multiplicities for cones. In addition, Monsky and his student Teixeira have found an exciting description in terms of what they call p-fractals that yields rationality of the Hilbert-Kunz multiplicity in many cases. Monsky has continued to prove new results on rationality, and this area is in constant development.

Mixed characteristic and the homological conjectures.

A few years ago a major step forward was made in the homological conjectures for rings of mixed characteristic with Heitmann's proof of the Direct Summand Conjecture in dimension three. This was a major advance, and it showed that many of the ideas from the positive characteristic case carry over and can be used successfully.

The new methods also have counterparts from an unexpected source, that of Arithmetic Geometry. The results outlined in the previous section have also been studied by Faltings, Gabber, Ramero, and others in their work on almost etale extensions with different aims, but with some very similar results, including a result on the vanishing of local cohomology similar to that of Heitmann. While there has been considerable progress in this area, there are still many facets of this connection that remain to be explored. One of the ideas from Arithmetic Geometry that has only recently been brought into this

subject is the use of a method of Fontaine to reduce questions in mixed characteristic to positive characteristic.

2. Integral dependence and integral closures.

The concepts of integral extensions and integral closures of rings are central to much of Commutative Algebra. In this part of the proposal we discuss an extension of these concepts to ideals. This defines a

closure operation for which the closure is in general larger than the tight closure discussed in the first section. Integral closure of ideals is closely related to singularity theory, the study of Rees algebras, and multiplier ideals.

Multiplier ideals and cores.

Multiplier ideals are integrally closed ideals that have been defined in complex algebraic geometry using log resolutions. Multiplier ideals play a role in the study of singularities, and more recently they have been related to arc spaces in the work of Ein, Mustata and their collaborators. The theory of multiplier ideals has surprising applications in local commutative algebra, as shown by Ein, Lazarsfeld and Karen Smith. On the other hand, the Japanese school of commutative algebraists was able to established a connection to test ideals, a characteristic p notion arising in tight closure theory. The proposed workshop will help to bring together the different points of view in this rich subject.

Multiplier ideals are also related to cores of ideals, a topic receiving a great deal of attention lately. A better understanding of cores would lead to improved versions of the celebrated Briancon-Skoda Theorem and solve a conjecture of Kawamata on the non-vanishing of sections of line bundles. This surprising connection was recently discovered by Smith and her coauthors, and one can expect further exciting developments in the near future.

Rees algebras and singularities.

There has been an abundance of new results about Rees algebras and their structure over the past fifteen years. Rees algebras are the rings in which integral dependence of ideals can be studied and they are the algebraic objects that appear in the process of resolution of singularities. The Cohen-Macaulay property of Rees algebras is of central importance, and has been studied extensively by Goto, Huneke, Ulrich, Vasconcelos and their collaborators. Rees algebras have been used in Kawasaki's celebrated proof of the existence of Macaulifications, a weak form of resolution of singularities. Another important contribution in singularity theory is Cutkosky's work on Abhyankar's conjecture about local factorization of birational maps between nonsingular varieties.

Rees algebras, in particular Rees algebras of modules, are the algebraic environment for geometric constructions like tangential varieties, Gauss images and secant varieties, the topic of the next section. The defining equations as well as the dimension of theses varieties can be obtained from a presentation of a suitable Rees algebra.

The concept of integral dependence of ideals and modules is essential in intersection theory and the theory of multiplicities. It plays an important role in equisingularity theory as well, through the work of

Gaffney, Kleiman and Teissier. On the other hand there is still no efficient algorithm for computing integral closures of ideals and modules, other than the wasteful approach through finding the integral

closure of the entire Rees algebra. So far this computational problem has evaded the efforts of many researchers. However, Vasconcelos and his coauthors have made considerable progress in proving bounds for the computational complexity of integral closures, based on generalized notions of multiplicities.

3. Secant varieties and Algebraic Statistics.

In the past few years we have seen some unforeseen applications of Commutative Algebra to many not obviously related fields. For example, the following questions have been considered. What is the fastest way to multiply two elements of a finite dimensional algebra? In particular, what is the best way to program matrix multiplication? What are efficient methods for dealing with massive data sets (e.g. results from clinical trials in cancer research, DNA sequences in human and animal genomes)? These seem like wildly different questions, but both are suprisingly connected to the study of questions in

Commutative Algebra that arise from Algebraic Geometry.

Secant varieties of Segre varieties.

In recent years many new methods and ideas have been developed to try and answer old, and unsolved, problems about the secant varieties of Segre varieties. Much of the impetus for studying these old problems came from the realization that there would be exciting applications for solutions. Thus, we are proposing to bring together people working on these problems in different ways and have them talk

about their methods.

The classical problems concern the dimensions of these varieties. But, new research has brought into focus the importance of understanding the defining equations of these varieties as well and these are the first steps, we now understand, in finding the entire minimal free resolution of the defining ideal. Also, the singularities of these varieties are a topic that has barely begun.

There have been different approaches to these problems. The group headed by Chiantini and Ciliberto (and mostly centered in Italy) has reexamined the work of the Italian `masters' of the late 19th and early 20th century on these issues. They have recast the main classical results in modern terms and pushed them to unforseen levels. This reexamination of the past has brought some splendid new ideas into the spotlight and the Workshop would be a perfect opportunity to have these ideas explained and discussed by experts.

The group headed by Catalisano, Geramita and Gimigliano has developed the pioneering work of Alexander and Hirshowitz in showing the connection between such problems and the study of non-reduced subschemes of projective space supported on unions of linear varieties. Their work on this approach has greatly expanded the collections of varieties for which the dimensions of the secant varieties are now known. They have identified certain key cases which need further study and the exposition of this research at the Workshop would provide a perfect opportunity to understand the thorny problems in this area.

Applications to Statistics.

The group in Algebraic Statistics, represented by Stillman-Sturmfels, has brought the problem of `defining equations' into focus with a provocative conjecture. A special case of this conjecture was recently settled by Landsberg using Lie Algebra techniques. Further special cases of an extension of the conjecture were also just provided by Catalisano-Geramita-Gimigliano. This area of study is brand new and offers opportunities for dramatic advances if considered by the appropriate researchers. We think that those people will be present at the Workshop.

Thus, the Workshop would provide the first opportunity to bring together these various groups. Also, since both the language of the modern understanding of these problems and the language of the conjectures which seek to solve them is the language of Commutative Algebra, a workshop which also includes experts on other aspects of the field (such as Rees algebras) would be ideal to provide the mix that offers the best chance for solving some of these questions.

For rings of prime characteristic, the Frobenius map, which sends an element of the ring to its pth power, where p is the characteristic of the ring, is an extremely powerful tool. Many conjectures on the

homological theory of Noetherian rings have been proven using this technique, and more recently these ideas have been extended in the theory of tight closure. For rings of mixed characteristic, an

important line of research has been to prove these conjectures and to extend these ideas to that case.

Positive characteristic, tight closure, and Hilbert-Kunz multiplicities.

The theory of tight closure was introduced by Hochster and Huneke a number of years ago and is still a very active area. Connections have been proven to exist between the concepts which arise naturally

in this theory and classical properties of singularities by Karen Smith, Watanabe, and others, and there is now a considerable amount of interest in relations with deep properties of local cohomology. Among the current topics of research in this area are test ideals, preservation under base change, and relations with local cohomology, including work by Smith, Lyubeznik, and Singh. A recent development has been the relation of the action of the Frobenius map on local cohomology to connectedness theorems. Another topic is the connection with multiplier ideals, an issue we discuss in more detail in the second section. Recently there has been a lot of progress on multiplier ideals, arc spaces and valuations in work of Ein, Mustata, and others; the connections with valuation theory are a classical topic that runs

through all areas of this subject.

The other topic of this section is Hilbert-Kunz multiplicities. These invariants are an analogue of traditional multiplicities in Algebraic Geometry, which are discussed in the second section of this proposal, and they record the action of the Frobenius homomorphism on a ring in positive characteristic. They are related to local Chern classes and other arithmetic invariants. Recent results by Brenner and Trivedi use results on vector bundles to compute these multiplicities for cones. In addition, Monsky and his student Teixeira have found an exciting description in terms of what they call p-fractals that yields rationality of the Hilbert-Kunz multiplicity in many cases. Monsky has continued to prove new results on rationality, and this area is in constant development.

Mixed characteristic and the homological conjectures.

A few years ago a major step forward was made in the homological conjectures for rings of mixed characteristic with Heitmann's proof of the Direct Summand Conjecture in dimension three. This was a major advance, and it showed that many of the ideas from the positive characteristic case carry over and can be used successfully.

The new methods also have counterparts from an unexpected source, that of Arithmetic Geometry. The results outlined in the previous section have also been studied by Faltings, Gabber, Ramero, and others in their work on almost etale extensions with different aims, but with some very similar results, including a result on the vanishing of local cohomology similar to that of Heitmann. While there has been considerable progress in this area, there are still many facets of this connection that remain to be explored. One of the ideas from Arithmetic Geometry that has only recently been brought into this

subject is the use of a method of Fontaine to reduce questions in mixed characteristic to positive characteristic.

2. Integral dependence and integral closures.

The concepts of integral extensions and integral closures of rings are central to much of Commutative Algebra. In this part of the proposal we discuss an extension of these concepts to ideals. This defines a

closure operation for which the closure is in general larger than the tight closure discussed in the first section. Integral closure of ideals is closely related to singularity theory, the study of Rees algebras, and multiplier ideals.

Multiplier ideals and cores.

Multiplier ideals are integrally closed ideals that have been defined in complex algebraic geometry using log resolutions. Multiplier ideals play a role in the study of singularities, and more recently they have been related to arc spaces in the work of Ein, Mustata and their collaborators. The theory of multiplier ideals has surprising applications in local commutative algebra, as shown by Ein, Lazarsfeld and Karen Smith. On the other hand, the Japanese school of commutative algebraists was able to established a connection to test ideals, a characteristic p notion arising in tight closure theory. The proposed workshop will help to bring together the different points of view in this rich subject.

Multiplier ideals are also related to cores of ideals, a topic receiving a great deal of attention lately. A better understanding of cores would lead to improved versions of the celebrated Briancon-Skoda Theorem and solve a conjecture of Kawamata on the non-vanishing of sections of line bundles. This surprising connection was recently discovered by Smith and her coauthors, and one can expect further exciting developments in the near future.

Rees algebras and singularities.

There has been an abundance of new results about Rees algebras and their structure over the past fifteen years. Rees algebras are the rings in which integral dependence of ideals can be studied and they are the algebraic objects that appear in the process of resolution of singularities. The Cohen-Macaulay property of Rees algebras is of central importance, and has been studied extensively by Goto, Huneke, Ulrich, Vasconcelos and their collaborators. Rees algebras have been used in Kawasaki's celebrated proof of the existence of Macaulifications, a weak form of resolution of singularities. Another important contribution in singularity theory is Cutkosky's work on Abhyankar's conjecture about local factorization of birational maps between nonsingular varieties.

Rees algebras, in particular Rees algebras of modules, are the algebraic environment for geometric constructions like tangential varieties, Gauss images and secant varieties, the topic of the next section. The defining equations as well as the dimension of theses varieties can be obtained from a presentation of a suitable Rees algebra.

The concept of integral dependence of ideals and modules is essential in intersection theory and the theory of multiplicities. It plays an important role in equisingularity theory as well, through the work of

Gaffney, Kleiman and Teissier. On the other hand there is still no efficient algorithm for computing integral closures of ideals and modules, other than the wasteful approach through finding the integral

closure of the entire Rees algebra. So far this computational problem has evaded the efforts of many researchers. However, Vasconcelos and his coauthors have made considerable progress in proving bounds for the computational complexity of integral closures, based on generalized notions of multiplicities.

3. Secant varieties and Algebraic Statistics.

In the past few years we have seen some unforeseen applications of Commutative Algebra to many not obviously related fields. For example, the following questions have been considered. What is the fastest way to multiply two elements of a finite dimensional algebra? In particular, what is the best way to program matrix multiplication? What are efficient methods for dealing with massive data sets (e.g. results from clinical trials in cancer research, DNA sequences in human and animal genomes)? These seem like wildly different questions, but both are suprisingly connected to the study of questions in

Commutative Algebra that arise from Algebraic Geometry.

Secant varieties of Segre varieties.

In recent years many new methods and ideas have been developed to try and answer old, and unsolved, problems about the secant varieties of Segre varieties. Much of the impetus for studying these old problems came from the realization that there would be exciting applications for solutions. Thus, we are proposing to bring together people working on these problems in different ways and have them talk

about their methods.

The classical problems concern the dimensions of these varieties. But, new research has brought into focus the importance of understanding the defining equations of these varieties as well and these are the first steps, we now understand, in finding the entire minimal free resolution of the defining ideal. Also, the singularities of these varieties are a topic that has barely begun.

There have been different approaches to these problems. The group headed by Chiantini and Ciliberto (and mostly centered in Italy) has reexamined the work of the Italian `masters' of the late 19th and early 20th century on these issues. They have recast the main classical results in modern terms and pushed them to unforseen levels. This reexamination of the past has brought some splendid new ideas into the spotlight and the Workshop would be a perfect opportunity to have these ideas explained and discussed by experts.

The group headed by Catalisano, Geramita and Gimigliano has developed the pioneering work of Alexander and Hirshowitz in showing the connection between such problems and the study of non-reduced subschemes of projective space supported on unions of linear varieties. Their work on this approach has greatly expanded the collections of varieties for which the dimensions of the secant varieties are now known. They have identified certain key cases which need further study and the exposition of this research at the Workshop would provide a perfect opportunity to understand the thorny problems in this area.

Applications to Statistics.

The group in Algebraic Statistics, represented by Stillman-Sturmfels, has brought the problem of `defining equations' into focus with a provocative conjecture. A special case of this conjecture was recently settled by Landsberg using Lie Algebra techniques. Further special cases of an extension of the conjecture were also just provided by Catalisano-Geramita-Gimigliano. This area of study is brand new and offers opportunities for dramatic advances if considered by the appropriate researchers. We think that those people will be present at the Workshop.

Thus, the Workshop would provide the first opportunity to bring together these various groups. Also, since both the language of the modern understanding of these problems and the language of the conjectures which seek to solve them is the language of Commutative Algebra, a workshop which also includes experts on other aspects of the field (such as Rees algebras) would be ideal to provide the mix that offers the best chance for solving some of these questions.