LESSON DIARY

03/03/2026 SPACES WITH FUNCTIONS

Definition of space with factions and of morphisms between them. Examples and first properties of spaces with functions. Definitions of affine (algebraic) variety, of (algebraic) variety and of morphism between two varieties. 

05/03/2026 EXAMPLES OF VARIETIES

Definition of the affine line and proof that it is an affine variety. Definition of the projective line.

06/03/2026 STRUCTURE THEOREM FOR AFFINE VARIETIES

Definition and properties of the projective line. Lists of assignments. Definition of the spectrum of a k-algebra, Spec(A), its topology and its structure of space with functions. Statement of the structure theorem: Spec(A) has a structure of affine variety such that A is isomorphic to k[Spec(A)]. Remarks. Spec(A) as the vanishing set of polynomials.  

10/03/2026 NOETHER'S LEMMA

Statement of Hilbert's Nullstellensatz. Noether's Lemma, with proof.  

12/03/2026 HILBERT'S NULLSTELLENSATZ

Lemma of Nakayama. Lemma: if A in B is an extension of k-algebras such that B is an A-module of finite type, then the restriction Spec(B) to Spec(A) is surjective. Review of the notion of rings of fractions and their main properties. Proof of Hilbet's Nullstellensatz.

17/03/2026 COROLLARIES OF THE NULLSTELLENSATZ

The NSS in the case where A=k[x_1, ..., x_n]. Affine algebraic sets. Corollary 1 of NSS: if the vanishing set of an ideal is empty, then 1 belongs to the ideal. Corollary 2 of the NSS: if D(f_l), for l in some index set L is an open cover of Spec(A), then 1 is in the ideal generated by the f_l's.  

18/03/2026 QUASI-COMPACTNESS OF SPECTRA

Corollary 3 of NSS: the spectrum of a f.g. k-algebra with 1 and without nilpotents is quasi-compact. Proof of the structure theorem of affine varieties: the map phi is an isomorphism.

19/03/2026 SUBVARIETIES

End of the proof of the Structure Theorem for affine varieties. Corollary: every variety is quasi-compact. Assignments. Definition of the induced structure of space with functions on a subset of a space with functions. Theorem/Definition: every locally closed subset of a variety X is a variety, called a subvariety of X. Examples: the affine n-space. Definition of quasi-affine varieties. 

20/03/2026 EXERCISES AND EXAMPLES

Discussion of the assigned exercises. Regular functions on the affine space minus a point. Definition of the projective n-space.

24/03/2026 PROJECTIVE SPACES

Definition of the projective spaces and proof that they are algebraic varieties. Definition of projective and quasi-projective varieties. Proposition: every closed subset of a projective space is the zero-set of homogenous polynomials. Definition of homogeneous ideals. Assignments. Definition of Noetherian rings. Hilbert's basis theorem (without proof): every commutative finitely generated k-algebra is Noetherian. 

25/03/2026 NOETHERIAN SPACES

Corollary of the Hilbert Basis Theorem: every open subset of a variety is quasi-compact. Definition of Noetherian topological space. Lemma: X is Noetherian if and only if every ascending chain of closed subsets becomes stationary. Definition of reducible and of irreducible topological space. Lemma: an affine variety X is irreducible if and only if k[X] is an integral domain.

26/03/2026 IRREDUCIBLE COMPONENTS

Definition of an (irreducible) component of a topological space. Proposition: a Noetherian topological space has finitely many components and their union is the whole space. Algebraic characterization of the components of an affine variety: they correspond to the minimal prime ideals of the ring of regular functions. Definition of the dimension of a topological space. 

27/03/2026 AFFINE AND FINITE MORPHISMS

Discussion of the assigned exercises. First properties of the concept of dimension and examples. Definition of affine and of finite morphisms. Examples.

31/03/2026 PROPERTIES OF FINITE MORPHISMS

The inclusion of a fine sub variety is a finite morphism. The inclusion of D(f) is an affine morphism. Proposition 1: Any finite morphism is a closed map. Proposition 2: if f: X to Y is finite, Z and W are closed subsets of X such that W is irreducible and Z is strictly contained in W, then f(Z) is strictly contained in f(W). Corollary 1: every fibre of a finite morphism is a finite set. 

01/04/2026 DIMENSION OF AFFINE SPACES

Corollary 2: Let f: X -> Y be a finite and subjective morphisms, then dim(X)=dim(Y). Theorem 1: dim (A^n)=n.

02/04/2026 HYPERSURFACES

Corollary 3: the dimension of the vanishing set of a non-constant polynomial in n variables is n-1. Proposition 3: the dimension of a variety is the maximum of the dimensions of the open sets of a cover. Corollary 4: the dimension of P^n is n. Corollary 5: every variety has finite dimension. Definition of hypersurface (in the affine n-space). Theorem 1: description of the components of a hypersurface; the k-algebra of regular functions of a hypersurface; every closed subset of A^n whose components are all of dimension n-1 is a hypersurface. 

09/04/2026 PRINCIPAL IDEAL THEOREM

Statement of the Principal Ideal Theorem (PIT). Lemma 1: the PIT in the affine case. Lemma 2: every open not empty subset of an irreducible variety has the same dimension of the variety.

10/04/2026 PROOF OF THE PRINCIPAL IDEAL THEOREM

Completion of the proof of Lemma 2. Proof of the Principal Ideal Theorem. Discussion of assignments. 

14/04/2026 PRODUCTS

Application of the Principal Ideal Theorem on the dimension of closed irreducible subsets in chains which are not refinable. Definition and existence of the product of two spaces with functions. Theorem: the product of two varieties is a variety. Definition of the tensor product of two modules over a ring. 

15/04/2026 TENSOR PRODUCT

Definition of the tensor product of two modules. Universal property. Dimension of the tensor product of two finite dimensional vector space. The structure of k-algebra on the tensor product of two k-algebras. 

16/04/2026 SEGRE EMBEDDING THEOREM

Universal Property 2, for the tensor product of two k-algebras. Proof of the theorem that says: the product of two varieties is a variety; if both varieties are affine, their product is also affine. Assignments. Statement of the Segre embedding theorem. 

17/04/2026 PROOF OF SEGRE EMBEDDING AND EXERCISES

Proof of the Segre embedding theorem. Corollary: the product of quasi-projective varieties is quasi-projective. Discussion of assignments. 

21/04/2026 SEPARATEDNESS

Definition of separated varieties, respectively spaces with functions. Example of non separated space: the line with two origins. Lemma 1: the graph of a morphism to a separated space is closed; a subspace of a separated space is separated; the product of two separated spaces is separated; quasi-affine and quasi-projective varieties are separated. Lemma 2: the graph of every morphism of varieties is locally closed. Proposition: a variety is separated if and only if the intersection of any pair of open affine sub varieties is affine and the regular functions are the restriction of regular functions on the open subsets. 

22/04/2026 PROJECTIVE NULLSTELLENSATZ

Relation between homogeneous ideals and cones with vertex in the origin. Statement and proof of the projective Nullstellensatz.

23/04/2026 DIMENSION AND INTERSECTION 

Proposition: the dimension of a cone is equal to the dimension of its image in the projective space plus one. Theorem: if two closed and irreducible sub varieties X and Y of the affine n-space intersect not trivially then each component of the intersection has dimension grater than or equal to dim(X)+dim(Y)-n. Theorem: if X and Y are two irreducible closed sub varieties of the projective n-space such that dim(X)+dim(Y)-n>=0, then X and Y have not trivial intersection. Corollary: the product of two projective spaces is not isomorphic to a projective space. 

24/04/2026 COMPLETENESS

Theorem: two irreducible closed sub varieties of the projective n-space whose sum of the dimension is >=n intersect not trivially. Definition of comple variety. First properties of complete varieties. Main Theorem of Elimination Theory.

28/04/2026 THE VERONESE EMBEDDING

End of the proof of the Main Theorem of Elimination Theory. Statement of Chow's Lemma. Definition and first properties of the Veronese Embedding. 

29/04/2026 IRREDUCIBILITY AND DIMENSION OF PRODUCTS

Discussion of the assignments: if Y is a sub variety of X and X is irreducible, then X=Y if and only if they have the same dimension; if X and Y are irreducible, then XxY is irreducible; dim(XxY)=dim(X)+dim(Y).

30/04/2026 PROJECTIONS FROM POINTS

Further properties of the Veronese Embedding. Assignments. Definition of the projection from a point in the projective space. Proposition: a projection from a point gives, via restriction, a finite morphism from a projective variety to its image.

05/05/2026 NOETHER NORMALIZATION THEOREM

Projections from linear subspaces and their interpretation as composition of projections from points. Statement and proof of Noether Normalization Theorem. Examples. 

07/05/2026 BLOW-UP

Definition and properties of the blow-up of the affine space at the origin. Definition and properties of the blow-up of the projective space at a point. Examples: strict transforms of plane curves. Definition of the field of rational functions of a irreducible variety.

08/05/2026 RATIONAL MAPS

Description of the field of rational functions of irreducible affine varieties. Proposition: open not empty subsets of irreducible varieties have isomorphic field of rational functions. Definition of rational maps. Examples. Definition of dominant rational maps and of birational maps and morphisms, birationally equivalent varieties. Pull-back of rational functions under dominant rational maps. 

12/05/2026 BIRATIONAL EQUIVALENT VARIETIES

Proposition: birational equivalent varieties have isomorphic fields of rational functions. Correspondence between dominant rational maps and homomorphisms of the fields of rational functions. Corollary: characterization of birational equivalent varieties. Corollary: birational equivalent varieties have the same dimension. Examples. Transcendence degree of finitely generated fields extension. The dimension of a variety coincides with the transcendence degree of its field of rational functions over the base field. Every variety is binational equivalent to a hyper surface.  

14/05/2026 ZARISKI TANGENT SPACES

Definition and properties of the local ring of a variety at a point. Local rings of an affine variety. Relations between the local rings of an irreducible variety and the field of rational functions. Definition of the Zariski tangent space of a variety at a point. The differential of a morphism at a point. Functorial properties of the differential. Isomorphic varieties have isomorphic tangent spaces. Description of the Zariski tangent space of an affine variety as the intersection of the kernels of the differentials of functions in the ideal of the variety. 

15/05/2026 ZARISKI TANGENT SPACES OF AFFINE VARIETIES

Proof of the description of the Zariski tangent space of an affine variety, at a point, as the intersection of the kernels of the differentials of functions in the ideal of the variety. Description of the differential of a morphism in terms of a Jacobi matrix of the morphism. Description of the Zariski tangent space of an affine variety as the intersection of the kernels of the differentials of functions that generate the ideal of the variety.

19/05/2026 SMOOTH POINTS

Taylor expansion of a polynomial at a point. The set of smooth points in an irreducible affine hyper surface is open and dense. Theorem: the set of smooth points in a irreducible variety is open and dense; at every point of a irreducible variety the dimension of the Zariski tangent space is greater or equal than the dimension of the variety. Criteria for smoothness of points in affine and projective varieties.

20/05/2026 SMOOTH CURVES

Discussion of two exercises. Definition of a curve. Definition of a Discrete Valuation Ring (DVR). Theorem: the local ring of a curve at a smooth point is a DVR. Lemma: corollary of Nakayama's lemma. Definition of the discrete valuation associated to the local ring of a smooth point of a curve.

21/05/2026 DIVISORS

Theorem: every rational map from a smooth curve to a projective variety extends to a morphism. Corollary: two smooth projective curves are birational equivalent if and only if they are isomorphic. Definition of the group of divisors and of the Picard group. The degree of a dominant rational map between curves. Pull-back of divisors. 

22/05/2026 RIEMANN-ROCH THEOREM

Principal divisors have degree 0. Definition of Pic^0. Proposition: a smooth projective curve C is isomorphic to P^1 if and only if Pic^0(C)=0.  Definition of the canonical divisor. Statement of the Riemann-Roch theorem and some consequences.