Attività settimanale
Introduzione
GEOMETRIA ALGEBRICA
anno accademico 2023/24
DOCENTE: Prof. Alberto ALBANO
email: alberto.albano@unito.it
tel.: 011 670 2890
INFORMAZIONI GENERALI
Il corso vale 6 Crediti (48 ore di lezione) e si svolge nel SECONDO semestre
ORARIO
- MAR/TUE 12:30 - 14:30
- GIO/THU 10:30 - 12:30
LEZIONI
Le lezioni verranno tutte svolte in Aula 5 (Palazzo Campana, secondo piano).
All classes will be held in Aula 5 (Palazzo Campana, second floor).
RICEVIMENTO DOCENTI / OFFICE HOURS
In generale, su appuntamento da concordare a lezione o via email.
By appointment. Ask in class or send an email.
PAGINA CAMPUSNET
ESAMI
L'esame consiste in una prova orale
Le date degli esami verranno concordate con gli studenti. Ci saranno 2 date nella sessione estiva (giugno/luglio 2024), una data nella sessione di settembre 2024 e due date nella sessione invernale (gennaio/febbraio 2025).
EXAMS
Final exam will be an oral exam
Exam dates will be fixed after consultation with the students. There will be 2 dates in the summer session (June/July 2024), one date in the fall session (September 2024) and 2 dates in the winter session (January/February 2025).
PROGRAMMA
Il corso si propone di illustrare alcuni aspetti di geometria algebrica, principalmente nel caso delle varietà proiettive. Seguiremo le note di Enrique Arrondo Introduction to projective varieties, scaricabili dalla pagina Enrique Arrondo - Elenco delle Pubblicazioni al fondo della lista, nella sezione Note. In questa pagina si possono trovare altre interessanti dispense su vari aspetti della Geometria Algebrica.
Gli argomenti sono simili a quelli trattati nel capitolo I (sezioni 1-5) del libro Algebraic Geometry di Robin Hartshorne e nel libro Basic Algebraic Geometry, vol. 1: Varieties in Projective Space di Igor' Šafarevič (Игорь Шафаревич) ma trattati in un ordine diverso e con un diverso uso dell'algebra commutativa per fondare rigorosamente la materia.
Sarà data per acquisita la parte introduttiva di geometria algebrica svolta in Istituzioni di Geometria.
Un programma più dettagliato si trova sulla pagina Campusnet.
PROGRAM
The course will be an introduction to the properties of projective algebraic varieties, following the notes of Enrique Arrondo Introduction to projective varieties, downloadable from the page Enrique Arrondo - List of Publications (look at the end of the list, under Lecture Notes). You can also find other interesting texts about various aspects of algebraic geometry.
ALTRI TESTI UTILI / USEFUL BOOKS
Geometria Algebrica / Algebraic Geometry:
- Miles Reid, Undergraduate Algebraic Geometry, London Mathematical Society, 1989.
- Joe Harris, Algebraic Geometry: A First Course, Springer, 1992.
- Klaus Hulek, Elementary Algebraic Geometry, American Mathematical Society, 2003.
- William Fulton, Algebraic Curves, scaricabile gratuitamente dal sito dell'autore / downloadable from Fulton's website.
Algebra Commutativa / Commutative Algebra:
- Serge Lang, Algebra, Springer 2002
- Michael F. Atiyah, Ian G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley 1969 (in biblioteca c'è anche una ristampa del 2019 / in the library there are several copies)
- Miles Reid, Undergraduate Commutative Algebra, Cambridge University Press, 1995, (evidenzia le relazioni con la geometria algebrica / written with applications to algebraic geometry in mind)
- David Eisenbud, Commutative algebra with a view toward algebraic geometry, Springer 1995
- Ernst Kunz, Introduction to commutative algebra and algebraic geometry, Birkhäuser, 1985
- Hideyuki Matsumura, Commutative algebra, Benjamin, 1970
- Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, 1986, sostanzialmente una versione aggiornata del libro del 1970 / basically an updated version of the 1970 book above
26 febbraio - 3 marzo
LEZIONE 1 -- martedì 27 febbraio 2024 -- ore 12:30 - 14:30
Algebra: basic facts on noetherian rings
- Hilbert Basis Theorem (Hilbertscher Basissatz)
- The Nullstellensatz (Hilbertscher Nullstellensatz)
For the Basis Theorem (and most algebraic prerequisites needed for this class), see Chapter 0 of Arrondo's notes
The proof of the Weak Projective Nullstellensatz seen in class is in Theorem 1.24.
LEZIONE 2 -- giovedì 29 febbraio 2024 -- ore 10:30 - 12:30
Resultant of two polynomials. Basic properties.
For more on resultants and a (different) proof of the Nullstellensatz see lecture of 1 october by J. Buhler Resultants, Discriminants, ...
Last part of the proof of the Weak Projective Nullstellensatz using resultants.
Affine Nullstellensatz:
- Weak: immediate from Weak Projective (see Arrondo, Theorem 14.2)
- Strong: using Rabinowitsch's trick (see Arrondo, Theorem 14.4)
Grassmannians: definition and Plucker embedding.
4 marzo - 10 marzo
LEZIONE 3 -- martedì 5 marzo 2024 -- ore 12:30 - 14:30
Grassmannians (Arrondo, Example 1.16, Exercises 1.17, 1.18): Plücker map; the Plücker map is well defined and injective; equations for the image.
Example: the Grassmannian \(\mathbf{G}(1, 3)\) is a quadric hypersurface in \(\mathbf{P}^5\).
Intersections and ideals (Arrondo, Example 1.21): non radical ideals for intersections contain more information than just the intersection of zero sets.
Decomposition of ideals:
- (Arrondo, Lemma 2.4) Every homogeneous ideal is the intersection of a finite number of irreducible ideals.
- (Arrondo, Lemma 2.5) An irreducible ideal is primary.
LEZIONE 4 -- giovedì 7 marzo 2024 -- ore 10:30 - 12:30
Primary decomposition of homogeneous ideals (Arrondo, Theorem 2.7). Statement and discussion of the meaning of uniqueness.
Example of an ideal with embedded primes and geometrical interpretation: \(I = (X_2^2, X_1 X_2) \)
Exercise: read Arrondo Example 1.23 and Exercise 2.8 to see another example of embedded components.
Hilbert function of an ideal and of a projective set
Hilbert functions of sets of 1, 2, 3 points and of a double point.
Lemma 3.2: the only ideals with Hilbert function definitely zero are the ones whose radical contains the irrelevant maximal ideal.
11 marzo - 17 marzo
LEZIONE 5 -- martedì 12 marzo 2024 -- ore 12:30 - 14:30
Prop. 3.8 (Arrondo) On the Hilbert functon of finite sets of points.
Examples: Hilbert function of \(\mathbf{P}^n\) and of the Veronese embedding of degree \(d\).
Exercise: find the Hilbert function of the Segre embedding.
A couple of exact sequences: Lemma 3.1 and Lemma 3.12.
LEZIONE 6 -- giovedì 14 marzo 2024 -- ore 10:30 - 12:30
Some more commutative algebra: Saturation of an ideal (Arrondo, Lemma 2.11 and Proposition 2.12)
Theorem 3.13. Existence (and uniqueness) of the Hilbert polynomial of a homogeneous ideal.
Lemma 3.16. Degree of the Hilbert polynomial and intersection with linear subspaces.
Theorem 3.17. Projective Hilbert's Nullstellensatz. The proof uses the Hilbert polynomial and the Weak Nullstellensatz (reformulated as Theorem 3.15 (i) using the Hilbert polynomial).
18 marzo - 24 marzo
LEZIONE 7 -- martedì 19 marzo 2024 -- ore 12:30 - 14:30
Definition of dimension for a projective set as the degree of the Hilbert polynomial.
Various examples: \(\mathbf{P}^n\), projective sets given by 1 equation.
Main properties of the dimension: Proposition 5.7. Discussion and proof of the various statements.
Proposition 5.9. For a projective set: codimension 1 if and only if given globally by 1 equation.
Definition of dimension of quasi projective sets. Proposition 5.8: discussion of the statement and overview of the proof. Read the details of the proof in Arrondo's notes.
LEZIONE 8 -- giovedì 21 marzo 2024 -- ore 10:30 - 12:30
Definition of the degree of a projective set via the leading coefficient of the Hilbert polynomial.
Examples: the degree of a hypersurface, the degree of a linear subspace.
Behaviour of the degree with respect to the union of projective sets.
Intersection multiplicity of projective sets when they meet in a point.
Theorem 5.15. The degree of a projective set \(X\) is the sum of the intersection multiplicities of the points in the intersection with a linear subspace of complementary dimension (assuming all the components of the intersection are points).
25 marzo - 31 marzo
LEZIONE 9 -- martedì 26 marzo 2024 -- ore 12:30 - 14:30
Bézout's theorem for plane curves.
Max Noether's \(AF + BG\) theorem.
Cayley-Bacharach theorem.
Application: associativity of the group law on a smooth plane cubic curve. For more details on the group law, see for example the book of Miles Reid "Undegraduate Algebraic Geometry".
Definition of the arithmetic genus and computation for plane curves.
Easter Homework:
- Read Proposition 2.15 on Arrondo's notes about the saturation of the ideal \((F,G)\) (needed for the proof of Noeher's theorem).
- Read Chapter 6 on Arrondo's notes about product and bihomogeneous polynomials/ideals.
We will discuss these topics, if needed, on Thursday 4 April.
VACANZA (PASQUA) -- giovedì 29 marzo 2024
1 aprile - 7 aprile
VACANZA (PASQUA) -- martedì 2 aprile 2024
LEZIONE 10 -- giovedì 4 aprile 2024 -- ore 10:30 - 12:30
Regular maps: definition, examples, isomorphisms, not every bijective morphism is an isomorphism.
First properties: Lemma 7.8. Discussion (mostly done in Istituzioni), proof of point (iii) about extensions of a morphism from a quasi projective to a projective domain.
Important example 7.9: blow up of a point in the plane.
8 aprile - 14 aprile
NO CLASS -- martedì 9 aprile 2024
LEZIONE 11 -- giovedì 11 aprile 2024 -- ore 10:30 - 12:30
First properties of regular maps \(f : X \to Y\), with \(X, Y\) projective:
- Theorem 7.11: the image of a projective set is closed
- Theorem 7.18: regular maps preserve irreducibility, do not increase dimension, the locus where the fibers have dimension greater or equal than \(k\) is closed in \(Y\)
15 aprile - 21 aprile
LEZIONE 12 -- martedì 16 aprile 2024 -- ore 12:30 - 14:30
Lemma 7.20 (Noether Normalizaton Lemma)
Theorem 7.21. If \(X\) is a projective variety of dimension \(r\), the every component of the intersection of \(X\) with a hypersurface not containing it has dimension \(r - 1\).
LEZIONE 13 -- giovedì 18 aprile 2024 -- ore 10:30 - 12:30
Properties of morphims with domain the projective space: Arrondo 8.1, 8.2.
Theorem 8.4: Dimension of the fibers of a morphism and irriducibility criterium.
22 aprile - 28 aprile
LEZIONE 14 -- martedì 23 aprile 2024 -- ore 12:30 - 14:30
Families of varieties. Definition, universal families. The universal hyperplane, the universal hypersurfaces. Grassmannians as families of linear spaces inside a fixed projective space.
Grassmannians: incidence correspondence (the universal family of \(k\)-dimensional spaces in an \(n\)-dimensional space. Irreducibility and dimension of Grassmannians (Arrondo, exercise 8.5)
Lines on surfaces of degree \(d\) in \(\mathbf{P}^3\). Statement of the problem, incidence correspondence \(\Gamma \subseteq G \times \mathbf{P}^N\), irreducibility and dimension of \(\Gamma\), discussion of the (simple) cases \(d = 1\) and \(d = 2\).
References:
- for families, Harris, Lecture 4
- for lines on surface: Shafarevich, Chapter 1.6.4, pagg.77 on
VACANZA -- giovedì 25 aprile 2024
29 aprile - 5 maggio
LEZIONE 15 -- martedì 30 aprile 2024 -- ore 12:30 - 14:30
Lines on cubic surfaces. Example of a cubic surface containing a finite number of lines. Every cubic surface contains at least one line.
Smooth surfaces of arbitrary degree containing lines: Fermat surfaces and Klein surfaces.
LEZIONE 16 -- giovedì 2 maggio 2024 -- ore 10:30 - 12:30
Equation for the incidence correspondence \(\Gamma\), so that \(\Gamma\) is projective.
The proof seen in class is taken from Shafarevich Basic Algebraic Geometry, vol. 1: Varieties in Projective Space, 6.4, pag. 77
The 27 lines on a smooth cubic surface.
For this part, see Miles Reid, Undergraduate Algebraic Geometry, par. 7.
6 maggio - 12 maggio
LEZIONE 17 -- martedì 7 maggio 2024 -- ore 12:30 - 14:30
System of plane cubic curves through 6 points in general position. Map \(\mathbf{P}^2 \dashrightarrow \mathbf{P}^3 \) and morphism \( F : X \to \mathbf{P}^3 \) blowing up the 6 points. \(F(X)\) is a cubic surface.
Description of the lines on \(F(X)\) using special curves in \(\mathbf{P}^2\). Intersection of lines on the cubic surface.
Definition of the (projective) tangent space to a projective set \(X \subseteq \mathbf{P}^n \).
Equations for the tangent space, using gradients of elements of the ideal \(I(X)\).
LEZIONE 18 -- giovedì 9 maggio 2024 -- ore 10:30 - 12:30
The dimension of the tangent space at a point is at least the dimension of the components through that point.
Definition of scmooth and singular points.
Smoothness and rank of the Jacobian matrix. The singular points are a closed subset.
The set of smooth points is (open) and non empty, hence dense, if \(X\) is irreducible.
13 maggio - 19 maggio
LEZIONE 19 -- martedì 14 maggio 2024 -- ore 12:30 - 14:30
End of the proof that the set of smooth point is non empty.
Comments on the tangent cone. Definition of the tangent cone for an hypersurface.
Simple singularities of plane curves, blow-ups, total transform and strict transform.
LEZIONE 20 -- giovedì 16 maggio 2024 -- ore 10:30 - 12:30
Detailed computations of blow-ups of a double point and of a cusp on a plane curve. The strict transform is smooth. Intersection of the strict transform and the exceptional divisor. Tangent spaces.
Example of a birational map: the Cremona map on the plane blowing up three points. The inverse map. The configuration of the exceptional divisors in the blow-up.
20 maggio - 26 maggio
LEZIONE 21 -- martedì 21 maggio 2024 -- ore 12:30 - 14:30
From now on, we follow Shafarevich book Basic Algebraic Geometry, vol. 1.
Regular functions on affine algebraic sets. Review of known facts from Istituzioni, see Shaf. Chapter 1.2.2, 1.2.3.
Rational functions on irreducible affine algebraic sets (see Shaf. Chapter 1.3.2). Definition of the field of fractions. Representative of a rational function. Regular points for a rational function.
- Important facts:
- Sh. Theorem 1.7, pag. 36: a rational function regular at all points is a regular function (a polynomial).
- The set of regular points (domain of the function) is open and non empty (and hence dense, since \(X\) is irreducible).
- A rational function is given uniquely if it is given on an open dense set.
Rational maps on an irreducible algebraic set (see Shaf. Chapter 1.3.3). Maps from \(X\) to affine space, to another closed set \(Y\).
A rational map with dense image induces an inclusion of function fields.
Definition of birational maps, birational equivalence, isomorphism of function fields.
Theorem 1.8. Any affine closed algebraic set is birationally equivalent to an affine hypersurface. The proof requires Shaf. Proposition A7, pag. 288, which is a consequence of the primitive element theorem, see any book in algebra or here https://en.wikipedia.org/wiki/Primitive_element_theorem
Exercise for next thursday: read Shafarevich, Chapter 1.4.2, pages 46-48, with the definition of a regular map on a quasi-projective variety (same definition as given before in Arrondo).
LEZIONE 22 -- giovedì 23 maggio 2024 -- ore 10:30 - 12:30
Rational functions of projective and quasiprojective varieties (see Shaf. Chapter 1.4.3)
- Important facts:
- If \(U \subseteq X\) is an open dense subset then \(K(U) = K(X)\)
- Domain of definition of a rational function, points of regularity, ...: same as in the affine case.
- A dominant rational map \(f: X \to Y\) induces a field inclusion \(f^* : K(Y) \to K(X)\). Birational maps
- Sh. Proposition 1.1, pag. 51: \(X\) and \(Y\) are birational if and only if they contain isomorphic open subsets.
Local ring of a point (see Shaf. Chapter 2.1.1)
Brief review of local rings and the process of localization of a ring at a prime ideal.
Definition of the local ring \(\mathscr{O}_x\) at a point as a subset of \(K(X)\) and as a localization of the ring of regular function in an open affine neighborhood.
Definition of the local ring along a subvariety \(\mathscr{O}_{X,Y}\)
27 maggio - 2 giugno
LEZIONE 23 -- martedì 28 maggio 2024 -- ore 12:30 - 14:30
Local equation of a subvariety. Lemma/definition as in Shaf. Lemma, pag. 107
Theorem (Sh. theorem 2.11, pag. 107) The local ring \(\mathscr{O}_{X, x}\) is a UDF for any \(x \in X\) smooth point. (WITHOUT PROOF).
Theorem (Sh. theorem 2.10, pag. 107) An irreducible subvariety of codimension 1 is given locally by 1 equation in the neighborhood of a smooth point.
Rational maps with values in projective space: the locus where the map is not defined has codimension at least 2.
- Corollaries:
- a rational map from a smooth curve to a projective curve is a morphism
- a birational map from a smooth, projective curve to a smooth, projective curve is an isomorphism
Theorem: for \(X\) smooth, irreducible and \(f\) a rational, non regular function, the locus on which \(f\) is not defined is of pure codimension 1 (i.e., all components have codimension 1) (WITHOUT PROOF)
Definition of (Weil) divisors. Group operation. Effective divisors.
LEZIONE 24 -- giovedì 30 maggio 2024 -- ore 10:30 - 12:30
Divisor associated to a rational function: definition of multiplicity along a subvariety \(v_Y(f)\).
- Properties of divisors of rational functions:
- \( (f) \ge 0\) \(\iff\) \(f\) is regular
- \( (f) = 0\) \(\iff\) \(f\) is regular and never vanishing on \(X\)
- for \(X\) projective, \( (f) = 0\) \(\iff\) \(f\) non zero constant
- for \(X\) projective, \( (f) = (g)\) \(\iff\) \(f = \lambda g\) for a non zero constant \(\lambda\)
Subgroup \(P(X)\) of principal divisors. The divisor class group \(\textrm{Cl}(X) = \textrm{Div}(X)/P(X)\).
Linear equivalence: two divisor \(D_1\) and \(D_2\) are linearly (or rationally) equivalent if there exists a rational function \(f\) such that \( D_1 - D_2 = (f) \).
Examples of principal and non principal divisors. Degree of a divisor on a curve.
Some easy computations: \(\textrm{Cl}(\mathbf{A}^n) = \{0\}\), \(\textrm{Cl}(\mathbf{P}^n) = \mathbf{Z}\)