IST courses on Algebraic Geometry 

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.

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.

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.

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.

In the last 2 lectures I will:

1. introduce (higher) secant varieties, (weakly) defective varieties and recall Terracini’s lemma;
2. I will talk about the famous theorem of Severi which classifies 1- defective surfaces;
3. I will talk about terracini’s extension to higher secant surfaces;
4. I will prove a bound on the degree of secant varieties;
5. I will classify surfaces for which the bound is achieved, showing how all this is related to castelnuovo–enriques’ theorem above.

Parts 4 & 5 above is recent joint work with F. Russo.

Applications to interpolation theory. Matching formulas for a degenerating family of embedded curves. Reduction lemmas for linear systems of plane curves with base fat points; recent results on linear systems via degeneration methods.

In the last 2 lectures I will:

1. introduce (higher) secant varieties, (weakly) defective varieties and recall Terracini’s lemma;
2. I will talk about the famous theorem of Severi which classifies 1- defective surfaces;
3. I will talk about terracini’s extension to higher secant surfaces;
4. I will prove a bound on the degree of secant varieties;
5. I will classify surfaces for which the bound is achieved, showing how all this is related to castelnuovo–enriques’ theorem above.

Parts 4 & 5 above is recent joint work with F. Russo.

Applications to interpolation theory. Matching formulas for a degenerating family of embedded curves. Reduction lemmas for linear systems of plane curves with base fat points; recent results on linear systems via degeneration methods.

In the first three lectures I will thouch these themes:

1. generalities on divisors and linear systems on a surface;
2. (birational) classification of movable linear systems whose general curve is irreducible of low geometric genus (say ≤ 2);
3. generalities on adjoint linear systems, concentrating on conditions for their nefness. If I have time I may say the little I know about their base point freeness,birationality etc.
4. I will indicate the proof of a classical theorem of Castelnuovo-Enriques which bounds (under suitable conditions) the dimensions of a linear system of curves in terms of their geometric genus. I will talk about extensions of this theorem both classical (Castelnuovo himself) and more recent (Reid). If time will be left, I will discuss possible extensions to threefolds.

Embedded degenerations in projective space with special incidence on rational surfaces, Veronese surfaces, line bundles on degenerations, degenerations of surfaces to unions of planes, formulas for plurigenera and $K^2$ .

In the first two and a half lectures I will thouch these themes:

1. generalities on divisors and linear systems on a surface;
2. (birational) classification of movable linear systems whose general curve is irreducible of low geometric genus (say ≤ 2);
3. generalities on adjoint linear systems, concentrating on conditions for their nefness. If I have time I may say the little I know about their base point freeness,birationality etc.
4. I will indicate the proof of a classical theorem of Castelnuovo-Enriques which bounds (under suitable conditions) the dimensions of a linear system of curves in terms of their geometric genus. I will talk about extensions of this theorem both classical (Castelnuovo himself) and more recent (Reid). If time will be left, I will discuss possible extensions to threefolds.

Embedded degenerations in projective space with special incidence on rational surfaces, Veronese surfaces, line bundles on degenerations, degenerations of surfaces to unions of planes, formulas for plurigenera and $K^2$ .

In the first two and a half lectures I will thouch these themes:

1. generalities on divisors and linear systems on a surface;
2. (birational) classification of movable linear systems whose general curve is irreducible of low geometric genus (say ≤ 2);
3. generalities on adjoint linear systems, concentrating on conditions for their nefness. If I have time I may say the little I know about their base point freeness,birationality etc.
4. I will indicate the proof of a classical theorem of Castelnuovo-Enriques which bounds (under suitable conditions) the dimensions of a linear system of curves in terms of their geometric genus. I will talk about extensions of this theorem both classical (Castelnuovo himself) and more recent (Reid). If time will be left, I will discuss possible extensions to threefolds.

Semistable degenerations of curves and surfaces. Normal crossings, triple point formulas, components with multiplicity; examples of degenerations of rational and $K3$ surfaces.

1. Etale topology, functor of points of a scheme, algebraic spaces; moduli problems, fine and coarse moduli spaces. Groupoids, groupoids in a category, groupoid fibrations, stacks.
2. Definition of algebraic stack. Examples: quotient stacks, moduli stacks. Morphisms to an algebraic stack, algebraic stacks as a $2$-category; fibered products. Representable morphisms, charts. (Quasi)coherent sheaves on a stack. properties of stacks and moprphisms (smoothness, separatedness, properness, etc).
3. More examples: modular forms, moduli spaces of stable curves and maps, Witten and Gromov-Witten invariants. The special case of orbifolds.

AIM:

The purpose of this course is to introduce a technique that has proven useful, especially in the theory of algebraic surfaces, to construct interesting varieties and families of varieties and to test general conjectures.

CONTENTS:

Double covers. Finite group actions on complex manifolds, quotient singularities. Structure theorem for abelian covers. Natural deformations of abelian covers. Applications and examples.

1. Etale topology, functor of points of a scheme, algebraic spaces; moduli problems, fine and coarse moduli spaces. Groupoids, groupoids in a category, groupoid fibrations, stacks.
2. Definition of algebraic stack. Examples: quotient stacks, moduli stacks. Morphisms to an algebraic stack, algebraic stacks as a $2$-category; fibered products. Representable morphisms, charts. (Quasi)coherent sheaves on a stack. properties of stacks and moprphisms (smoothness, separatedness, properness, etc).
3. More examples: modular forms, moduli spaces of stable curves and maps, Witten and Gromov-Witten invariants. The special case of orbifolds.

AIM:

The purpose of this course is to introduce a technique that has proven useful, especially in the theory of algebraic surfaces, to construct interesting varieties and families of varieties and to test general conjectures.

CONTENTS:

Double covers. Finite group actions on complex manifolds, quotient singularities. Structure theorem for abelian covers. Natural deformations of abelian covers. Applications and examples.

1. Etale topology, functor of points of a scheme, algebraic spaces; moduli problems, fine and coarse moduli spaces. Groupoids, groupoids in a category, groupoid fibrations, stacks.
2. Definition of algebraic stack. Examples: quotient stacks, moduli stacks. Morphisms to an algebraic stack, algebraic stacks as a $2$-category; fibered products. Representable morphisms, charts. (Quasi)coherent sheaves on a stack. properties of stacks and moprphisms (smoothness, separatedness, properness, etc).
3. More examples: modular forms, moduli spaces of stable curves and maps, Witten and Gromov-Witten invariants. The special case of orbifolds.

AIM:

The purpose of this course is to introduce a technique that has proven useful, especially in the theory of algebraic surfaces, to construct interesting varieties and families of varieties and to test general conjectures.

CONTENTS:

Double covers. Finite group actions on complex manifolds, quotient singularities. Structure theorem for abelian covers. Natural deformations of abelian covers. Applications and examples.