IST courses on Algebraic Geometry 

04/02/2010, 14:00 — 15:00 — Room P3.10, Mathematics Building
Lucia Caporaso, Università di Roma III

Compactifications of Jacobians (IV)

See also

https://www.math.ist.utl.pt/~mmlopes/ISTCOURSES/

04/02/2010, 11:30 — 12:30 — Room P3.10, Mathematics Building
Eduardo Esteves, Instituto Nacional de Matemática Pura e Aplicada

Compactifications of Jacobians (IV)

See also

https://www.math.ist.utl.pt/~mmlopes/ISTCOURSES/

03/02/2010, 15:30 — 16:30 — Room P3.10, Mathematics Building
Lucia Caporaso, Università di Roma III

Compactifications of Jacobians (III)

See also

https://www.math.tecnico.ulisboa.pt/~mmlopes/ISTCOURSES/

03/02/2010, 14:00 — 15:00 — Room P3.10, Mathematics Building
Eduardo Esteves, Instituto Nacional de Matemática Pura e Aplicada

Compactifications of Jacobians (III)

See also

https://www.math.ist.utl.pt/~mmlopes/ISTCOURSES/

02/02/2010, 15:30 — 16:30 — Room P3.10, Mathematics Building
Lucia Caporaso, Università di Roma III

Compactifications of Jacobians (II)

See also

https://www.math.tecnico.ulisboa.pt/~mmlopes/ISTCOURSES/

02/02/2010, 14:00 — 15:00 — Room P3.10, Mathematics Building
Eduardo Esteves, Instituto Nacional de Matemática Pura e Aplicada

Compactifications of Jacobians (II)

See also

https://www.math.ist.utl.pt/~mmlopes/ISTCOURSES/

01/02/2010, 15:30 — 16:30 — Room P3.10, Mathematics Building
Lucia Caporaso, Università di Roma III

Compactifications of Jacobians (I)

Jacobians play a major role in the theory of smooth algebraic curves. For singular curves their analogues are not compact. We will present the theory of compactifications of Jacobians of singular curves, concentrating on nodal curves. Topics covered are: construction of compactifications, theta divisors, Abel maps, Brill-Noether theory and Torelli theorem.

See also

https://www.math.ist.utl.pt/~mmlopes/ISTCOURSES/

01/02/2010, 14:00 — 15:00 — Room P3.10, Mathematics Building
Eduardo Esteves, Instituto Nacional de Matemática Pura e Aplicada

Compactifications of Jacobians (I)

Jacobians play a major role in the theory of smooth algebraic curves. For singular curves their analogues are not compact. We will present the theory of compactifications of Jacobians of singular curves, concentrating on nodal curves. Topics covered are: construction of compactifications, theta divisors, Abel maps, Brill-Noether theory and Torelli theorem.

See also

https://www.math.ist.utl.pt/~mmlopes/ISTCOURSES/

10/09/2009, 14:30 — 15:30 — Room P3.10, Mathematics Building
Giuseppe Pareschi, Università di Roma II, Italia

Generic vanishing and continuous global generation on irregular and abelian varieties

Two basic tools provided by the theory of coherent sheaves on projective varieties are vanishing of the cohomology and global gen- eration. On irregular varieties there are wonderful natural weakenings of such notions: generic vanishing and continuous global generation. They go together with the systematic use of the Fourier-Mukai trans- form. These ideas have their roots in three important bodies of work:

• Mukai's theory of the Fourier-Mukai trasform;
• Green-Lazarsfeld's generic vanishing theorems, and their applica- tions to the geometry of irregular varieties due to Ein, Lazarsfeld, Hacon and others;
• Kempf's work on theta-functions. In my lectures I will focus on such concepts. Moreover I will describe several concrete applications to geometric and algebraic problems, as syzygies of abelian varieties, special subvarieties of abelian varieties, invariants and pluricanonical maps of irregular varieties of maximal Albanese dimension.

10/09/2009, 12:00 — 13:00 — Room P3.10, Mathematics Building
Mihnea Popa, University of Illinois at Chicago

Fourier-Mukai and BGG transforms in the cohomological study of projective varieties

The Fourier-Mukai transform is an equivalence between the derived categories of coherent sheaves on an abelian variety, and on its dual variety. Introduced by Mukai, it has found numerous applications and has become a fundamental tool in the study of abelian varieties and, more generally, of irregular varieties. Furthermore, similar “integral" transforms play a major role in studying the birational geometry of algebraic varieties via derived categories (as for example in the work of Bondal-Orlov, Bridgeland and Kawamata). The Bernstein-Gel'fand-Gel'fand (BGG) correspondence is an equivalence between the derived category of modules over the exterior algebra of a vector space and that of linear complexes of modules over the symmetric algebra of the dual vector space. It has been recently further developed in work of Eisenbud-Floystad-Schreyer.

In these lectures I will focus on applications of these two types of equivalences to the cohomological study of irregular varieties (or compact Kaehler manifolds). I will explain how they can be used to extend the Generic Vanishing theorems of Green-Lazarsfeld, and to bound the holomorphic Euler characteristic and the Hodge numbers of varieties without higher irrational pencils. I will also describe a surprisingly natural structure that the cohomology of the canonical bundle acquires as a module over the exterior algebra. Plenty of concrete applications will be provided.

09/09/2009, 16:00 — 17:00 — Room P3.10, Mathematics Building
Giuseppe Pareschi, Università di Roma II

Generic vanishing and continuous global generation on irregular and abelian varieties

Two basic tools provided by the theory of coherent sheaves on projective varieties are vanishing of the cohomology and global gen- eration. On irregular varieties there are wonderful natural weakenings of such notions: generic vanishing and continuous global generation. They go together with the systematic use of the Fourier-Mukai trans- form. These ideas have their roots in three important bodies of work:

• Mukai's theory of the Fourier-Mukai trasform;
• Green-Lazarsfeld's generic vanishing theorems, and their applica- tions to the geometry of irregular varieties due to Ein, Lazarsfeld, Hacon and others;
• Kempf's work on theta-functions. In my lectures I will focus on such concepts. Moreover I will describe several concrete applications to geometric and algebraic problems, as syzygies of abelian varieties, special subvarieties of abelian varieties, invariants and pluricanonical maps of irregular varieties of maximal Albanese dimension.

09/09/2009, 14:00 — 15:00 — Room P3.10, Mathematics Building
Mihnea Popa, University of Illinois at Chicago

Fourier-Mukai and BGG transforms in the cohomological study of projective varieties

The Fourier-Mukai transform is an equivalence between the derived categories of coherent sheaves on an abelian variety, and on its dual variety. Introduced by Mukai, it has found numerous applications and has become a fundamental tool in the study of abelian varieties and, more generally, of irregular varieties. Furthermore, similar "integral" transforms play a major role in studying the birational geometry of algebraic varieties via derived categories (as for example in the work of Bondal-Orlov, Bridgeland and Kawamata). The Bernstein-Gel'fand-Gel'fand (BGG) correspondence is an equivalence between the derived category of modules over the exterior algebra of a vector space and that of linear complexes of modules over the symmetric algebra of the dual vector space. It has been recently further developed in work of Eisenbud-Floystad-Schreyer.

In these lectures I will focus on applications of these two types of equivalences to the cohomological study of irregular varieties (or compact Kaehler manifolds). I will explain how they can be used to extend the Generic Vanishing theorems of Green-Lazarsfeld, and to bound the holomorphic Euler characteristic and the Hodge numbers of varieties without higher irrational pencils. I will also describe a surprisingly natural structure that the cohomology of the canonical bundle acquires as a module over the exterior algebra. Plenty of concrete applications will be provided.

08/09/2009, 16:00 — 17:00 — Room P3.10, Mathematics Building
Mihnea Popa, University of Illinois at Chicago

Fourier-Mukai and BGG transforms in the cohomological study of projective varieties

The Fourier-Mukai transform is an equivalence between the derived categories of coherent sheaves on an abelian variety, and on its dual variety. Introduced by Mukai, it has found numerous applications and has become a fundamental tool in the study of abelian varieties and, more generally, of irregular varieties. Furthermore, similar "integral" transforms play a major role in studying the birational geometry of algebraic varieties via derived categories (as for example in the work of Bondal-Orlov, Bridgeland and Kawamata). The Bernstein-Gel'fand-Gel'fand (BGG) correspondence is an equivalence between the derived category of modules over the exterior algebra of a vector space and that of linear complexes of modules over the symmetric algebra of the dual vector space. It has been recently further developed in work of Eisenbud-Floystad-Schreyer.

In these lectures I will focus on applications of these two types of equivalences to the cohomological study of irregular varieties (or compact Kaehler manifolds). I will explain how they can be used to extend the Generic Vanishing theorems of Green-Lazarsfeld, and to bound the holomorphic Euler characteristic and the Hodge numbers of varieties without higher irrational pencils. I will also describe a surprisingly natural structure that the cohomology of the canonical bundle acquires as a module over the exterior algebra. Plenty of concrete applications will be provided.

08/09/2009, 14:00 — 15:00 — Room P3.10, Mathematics Building
Giuseppe Pareschi, Università di Roma II, Italia

Generic vanishing and continuous global generation on irregular and abelian varieties

Two basic tools provided by the theory of coherent sheaves on projective varieties are vanishing of the cohomology and global gen- eration. On irregular varieties there are wonderful natural weakenings of such notions: generic vanishing and continuous global generation. They go together with the systematic use of the Fourier-Mukai trans- form. These ideas have their roots in three important bodies of work:

• Mukai's theory of the Fourier-Mukai trasform;
• Green-Lazarsfeld's generic vanishing theorems, and their applica- tions to the geometry of irregular varieties due to Ein, Lazarsfeld, Hacon and others;
• Kempf's work on theta-functions. In my lectures I will focus on such concepts. Moreover I will describe several concrete applications to geometric and algebraic problems, as syzygies of abelian varieties, special subvarieties of abelian varieties, invariants and pluricanonical maps of irregular varieties of maximal Albanese dimension.

07/09/2009, 16:00 — 17:00 — Room P3.10, Mathematics Building
Giuseppe Pareschi, Università di Roma II

Generic vanishing and continuous global generation on irregular and abelian varieties

Two basic tools provided by the theory of coherent sheaves on projective varieties are vanishing of the cohomology and global gen- eration. On irregular varieties there are wonderful natural weakenings of such notions: generic vanishing and continuous global generation. They go together with the systematic use of the Fourier-Mukai trans- form. These ideas have their roots in three important bodies of work:

• Mukai's theory of the Fourier-Mukai trasform;
• Green-Lazarsfeld's generic vanishing theorems, and their applica- tions to the geometry of irregular varieties due to Ein, Lazarsfeld, Hacon and others;
• Kempf's work on theta-functions. In my lectures I will focus on such concepts. Moreover I will describe several concrete applications to geometric and algebraic problems, as syzygies of abelian varieties, special subvarieties of abelian varieties, invariants and pluricanonical maps of irregular varieties of maximal Albanese dimension.

07/09/2009, 14:00 — 15:00 — Room P3.10, Mathematics Building
Mihnea Popa, University of Illinois at Chicago

Fourier-Mukai and BGG transforms in the cohomological study of projective varieties

The Fourier-Mukai transform is an equivalence between the derived categories of coherent sheaves on an abelian variety, and on its dual variety. Introduced by Mukai, it has found numerous applications and has become a fundamental tool in the study of abelian varieties and, more generally, of irregular varieties. Furthermore, similar "inte- gral" transforms play a major role in studying the birational geometry of algebraic varieties via derived categories (as for example in the work of Bondal-Orlov, Bridgeland and Kawamata). The Bernstein-Gel'fand-Gel'fand (BGG) correspondence is an equivalence between the derived category of modules over the exterior algebra of a vector space and that of linear complexes of modules over the symmetric algebra of the dual vector space. It has been recently further developed in work of Eisenbud-Floystad-Schreyer.

In these lectures I will focus on applications of these two types of equivalences to the cohomological study of irregular varieties (or com- pact Kaehler manifolds). I will explain how they can be used to extend the Generic Vanishing theorems of Green-Lazarsfeld, and to bound the holomorphic Euler characteristic and the Hodge numbers of varieties without higher irrational pencils. I will also describe a surprisingly nat- ural structure that the cohomology of the canonical bundle acquires as a module over the exterior algebra. Plenty of concrete applications will be provided.

18/05/2006, 15:45 — 16:45 — Room P3.10, Mathematics Building
Igor Dolgachev, University of Michigan

Introduction to classical Cremona transformations II

The group structure of the group $Cr(2)$ of Cremona transformations of projective plane is unlike any other familiar group structures. Although it is generated by projective transformations and a single nonprojective transformation, its structure is very complicated. For example, the conjugacy classes of elements of given finite order are parametrized by an algebraic variety with finitely many irreducible components of different dimension. This is very different from the case of Lie groups. One of the oldest conjectures is that the group is simple as an abstract group.

In these lectures we will briefly discuss the now completed classification of conjugacy classes of finite groups of $Cr(2)$ which is equivalent to the classification of pairs $(S,G)$, where $S$ is a rational surface and $G$ is a finite group of its automorphisms, up to equivariant birational maps.

We will also discuss known examples of infinite subgroups of $Cr(2)$ which can be realized as automorphism groups of rational surfaces, and their relationship to complex dynamics of rational maps.

18/05/2006, 14:30 — 15:30 — Room P3.10, Mathematics Building
Sandro Verra, Universitá di Roma III

Introduction to classical Cremona transformations I

A Cremona transformation is a birational automorphism of a complex projective space $\mathbb{P}^r$. The study of such transformations and the group which they generate was a popular subject of classical algebraic geometry flourishing more than one century ago. Although much progress has been made in the two-dimensional case, in spite of the efforts of many classical and modern algebraic geometers, most of the fundamental problems in the higher-dimensional case remain unsolved. The aim of these lectures is to report on the classical techniques for studying Cremona transformations and specifically on the rich legacy of classical examples. The plan is to present a series of examples in modern terms and to use these examples to introduce basic constructions and techniques, as well as more recent results and open problems. The program will include some of the following topics:

• Base locus and numerical characters of Cremona transformations.
• Transformations defined by quadrics.
• The cubo-cubic transformation of $\mathbb{P}^3$ and related topics.
• Birational involutions of $\mathbb{P}^2$ and $\mathbb{P}^3$.
• Homaloidal linear system of surfaces with finite base locus.
• Cremona transformations with smooth and connected base locus.
• Classification problems.

17/05/2006, 15:45 — 16:45 — Room P3.10, Mathematics Building
Igor Dolgachev, University of Michigan

Introduction to classical Cremona transformations II

The group structure of the group $Cr(2)$ of Cremona transformations of projective plane is unlike any other familiar group structures. Although it is generated by projective transformations and a single nonprojective transformation, its structure is very complicated. For example, the conjugacy classes of elements of given finite order are parametrized by an algebraic variety with finitely many irreducible components of different dimension. This is very different from the case of Lie groups. One of the oldest conjectures is that the group is simple as an abstract group.

In these lectures we will briefly discuss the now completed classification of conjugacy classes of finite groups of $Cr(2)$ which is equivalent to the classification of pairs $(S,G)$, where $S$ is a rational surface and $G$ is a finite group of its automorphisms, up to equivariant birational maps.

We will also discuss known examples of infinite subgroups of $Cr(2)$ which can be realized as automorphism groups of rational surfaces, and their relationship to complex dynamics of rational maps.

17/05/2006, 14:30 — 15:30 — Room P3.10, Mathematics Building
Sandro Verra, Universitá di Roma III

Introduction to classical Cremona transformations I

A Cremona transformation is a birational automorphism of a complex projective space $\mathbb{P}^r$. The study of such transformations and the group which they generate was a popular subject of classical algebraic geometry flourishing more than one century ago. Although much progress has been made in the two-dimensional case, in spite of the efforts of many classical and modern algebraic geometers, most of the fundamental problems in the higher-dimensional case remain unsolved. The aim of these lectures is to report on the classical techniques for studying Cremona transformations and specifically on the rich legacy of classical examples. The plan is to present a series of examples in modern terms and to use these examples to introduce basic constructions and techniques, as well as more recent results and open problems. The program will include some of the following topics:

• Base locus and numerical characters of Cremona transformations.
• Transformations defined by quadrics.
• The cubo-cubic transformation of $\mathbb{P}^3$ and related topics.
• Birational involutions of $\mathbb{P}^2$ and $\mathbb{P}^3$.
• Homaloidal linear system of surfaces with finite base locus.
• Cremona transformations with smooth and connected base locus.
• Classification problems.