## Weekly outline

### Geometria Algebrica

Laurea Magistrale in Matematica, secondo semestre, 48 ore, 6 crediti.

Docente:

__Cinzia Casagrande__*Schedule:*Tuesday 12:30-14:30, Friday 14:30-16:30, room 2*Office hours:*Thursday 11-12 in the professor's office, or on appointment**.**References:

Shafarevich, Basic Algebraic Geometry, vol. 1

Reid, Undergraduate Algebraic Geometry

Harris, Algebraic Geometry - A First Course

Hartshorne, Algebraic Geometry (cap. 1)Referenze di algebra:

Lang, Algebra

Atiyah and Macdonald, Introduction to Commutative Algebra**Important notices on the schedule:**

on Friday 7/4 the lecture will be at 10.30-12.30 (room 2)

on Tue 9/5, Fri 12/5 and Fri 26/5 the course will not take place

these 3 lectures will take place on: Mon 3/4, Mon 22/5, Mon 29/5 at 12.30-14.30 (room 2)### 26 February - 4 March

**Avviso importante: martedi' 28/2 non si terra' la lezione di Geometria Algebrica; il corso iniziera' venerdi' 3/3.****3/3, 10.30-12.30:**Introduction to the course, practical information. Algebraic subsets in affine space, Zariski topology on A^{n}, examples. Principal open subsets; the principal open subsets are a basis for the Zariski topology. Ideal of a closed subset, elementary properties. Hilbert Nullstellensatz, equivalent statements (no proof), discussion about the algebraic closure of the field. Applications. Noetherian topological spaces and irreducible topological spaces.Exercises: 1) S. (= from Shafarevich's book) ex. 1 p. 32 (end of Ch. I sec. 2). 2) Show that if k is a field where the Nullstellensatz holds, then k is algebraically closed. 3) Let k be alg. closed and f,g polynomials such that V(f)=V(g). Explain the relation among f and g.

### 5 March - 11 March

**7/3, 12.30-14.30:**Proof of existence and uniqueness of the decomposition in irreducible components for a noetherian topological space. Application to affine closed subsets. An affine closed subset X is irreducible if and only if I(X) is a prime ideal.

Affine hypersurfaces, irreducible components, degree. Singular and non-singular points, tangent space. Examples. Multiplicity of a singular point of a hypersurface. Singular points of multiplicity 2 of affine plane curves: nodes, cusps. If a plane curve X of degree d has a singular point p of multiplicity d, then X is a union of d lines through p. Tangent lines to a hypersurface X at a point p, a line is tangent to X at p iff it is contained in the tangent space T_{p}X. Principal tangent lines to a curve at a singular point, examples.Exercises: (1) Let X be a topological space, and Y a subset. Prove that Y is irreducible iff its closure is irreducible. (2) Prove that a closed subset in a noetherian topological space is noetherian. (3) Prove that a noetherian topological space is compact. (4) Prove that a noetherian topological space is Hausdorff iff it is finite. (5) S. ex.1 page 40 (end of Ch. I sec. 3).

**10/3, 14.30-16.30:**If X is an irreducible affine hypersurface, then Sing(X) is a proper closed subset; if X=X_{1}u...uX_{r}irreducible components, then Sing(X) is a closed subset given by the union of Sing(X_{i}) with the intersections among any two distinct components. If X is a complex algebraic hypersurface in A^{n}, the its regular locus has a natural structure of real differentiable manifold of dimension 2(n-1) (sketch of proof for an affine curve). k-algebras, finitely generated k-algebras. Regular functions on a closed subset X of A^{n}, ring of regular functions k[X]. Examples.Exercises: (1) Show that a regular function is continuous. (2) Show that given F in k[x,y], F(x,1/x)=0 in k(x) if and only if xy-1 divides F. (3) S. ex. 6 page 32 (end of Ch. I sec. 2).

### 12 March - 18 March

**14/3, 12.30-14.30:**Correspondence between closed subsets of X and ideals of k[X]. Regular maps between affine closed subsets, isomorphisms. Pull-back of a regular function under a morphism, homomorphism of k-algebras induced by a morphism, functoriality. Given a k-algebra homomorphism h:k[X]->k[X], there exists a unique morphism f:X->Y such that h=f^{*}. Consequences: f is an isomorphism iff f^{*}is an isomorphism; X and Y are isomorphic iff k[X] and k[Y] are isomorphic; there is a bijection between isomorphism classes of affine closed subsets, and isomorphism classes of finitely generated k-algebras with no nilpotents. Dominant morphisms; f is dominant iff f^{*}is injective. Examples: an isomorphism, a morphism which is not open nor closed, a bijective morphism which is not an isomorphism. The Frobenius map in positive characteristic. Automorphisms of the affine plane, the Jacobian conjecture.

Products of closed subsets; the Zariski topology in A^{n+m}is (strictly) finer than the product of the Zariski topologies in A^{n}and A^{m}. The diagonal of X is closed in XxX. The set where two morphisms are equal is closed.Exercises: (1) the composition of two morphisms is a morphism; (2) a morphism is continuous; (3) S. ex. 4, 5, 8, 12 pages 32-33.

**17/3, 14.30-16.30:**Discussion of the exercises assigned in the previous lectures.

Given X, Y closed subsets, I(XxY) is generated by I(X) and I(Y), and XxY is irreducible if and only if X and Y are irreducible. Decomposition in irreducible components of XxY in terms of the decompositions of X and Y. Structure of k-algebra on the tensor product (over k) of two k-algebras; k[XxY] is isomorphic to the tensor product of k[X] and k[Y].Exercises: (1) Let Z be a closed subset of XxY. Show that the set {x in X|{x}xY is contained in Z} is closed in X. (2) Let f:X->Y be a morphism. Show that the graph of f G={(x,f(x))} is closed in XxY, and that it is isomorphic to X. (3) Let X,Y be closed subsets of A

^{n}. Show that the intersection of X and Y is isomorphic to the intersection of XxY with the diagonal, in A^{2n}.### 19 March - 25 March

**21/3, 12.30-14.30:**Closed subsets of the projective space and Zariski topology, homogeneous ideals, correspondence ideals/closed subsets in the projective case, affine cone of a projective closed subset, projective Nullstellensatz. Affine charts on the projective space; the Zariski topology of P^{n}induces the affine Zariski topology on the affine charts; projective closure of an affine closed subset. Examples. Quasi-projective varieties. Every quasi-projective variety is a noetherian topological space and has a decomposition in irreducible components.

Projective transformations, projectively equivalent closed subsets. The ideal of two disjoint lines in P^{3}.Exercises: (1) Show that a quasi-projective variety is open in its projective closure. (2) Show that a quasi-projective variety contained in an affine chart is an open subset of an affine closed subset. (3) Show that open and closed subsets of a quasi-projective variety are quasi-projective varieties. (4) Show that an open subset of a noetherian topological space is again noetherian. (5) Let Y={(t,t

^{2},t^{3})} in A^{3}and let X be the projective closure of Y in P^{3}. Find homogeneous polynomial defining X. (6) Show that the irreducible factors of a homogeneous polynomial are homogeneous. (7) Show that the radical of a homogeneous ideal is homogeneous.**23/3, 14.30-16.30:**Projective hypersuperfaces. Examples: quadrics, rank of a quadric, standard form, plane conics. Parametrizations of degree d hypersurfaces in P^{n}by the projectivization of the vector space of homogeneous polynomials of degree d. Case d=1, dual projective space. Case of plane conics; relation with symmetrix matrices. Discriminant hypersurface D in P^{5}parametrizing conics of rank <3; closed locus Z in P^{5}parametrizing double lines. Projective tangent space to a hypersurface, homogeneous equation; singular and non-singular points of a projective hypersurface. Example: conics.

A non-singular complex projective plane curve, with the euclidean topology, is a compact orientable surface; hints on the genus formula.

Regular functions on a quasi-projective variety. The definition coincides with the previous one for closed affine subsets.Exercises: (1) Show that a quadric is irreducible if and only if it has rank >2. (2) Show that the set of hypersurfaces of degree d containing a fixed point is a hyperplane in P

^{N}. (3) Show that given a quadric Q, if f is the symmetric bilinear form on k^{n+1}associated to Q, then Sing(Q) is the projective subspace corresponding to ker(f). (4) Show that Z=Sing(D) in P^{5}. (5) Let C be a conic given by the union of two distinct lines L_{1}and L_{2}intersecting in p. Show that the projective tangent space to D at [C] is the hyperplane of conics through p.### 26 March - 1 April

**28/3, 12.30-14.30:**Every regular function on the projective space is constant. Regular functions on a principal open subset of an affine closed subset. Every regular function on the affine plane minus a point is regular on the whole plane. Regular functions are continuous. An example of a regular function on an open subset of an affine closed subset which cannot be given globally as a quotient of two polynomials. Regular maps between quasi projective varieties. Example: a morphism from the projective line to an irreducible conic and its inverse. If f:X->Y and Y is contained in an affine space, then f is regular if and only if the components of f are regular.Exercise: let n>1 and d>1. Give an example of a non singular hypersurface of degree d in P

^{n}. If (n,d) is different from (2,2), give an example of an irreducible singular hypersurface of degree d in P^{n}. What happens if n=1, d=1, or (n,d)=(2,2)?**31/3, 14.30-16.30:**Example: the twisted cubic. Isomorphisms. Projectively equivalent varieties in P^{n}are isomorphic. The composition of two morphisms is a morphism. A morphism is continuous. Pull-back of regular functions under a morphism; functoriality. Principal open subsets of a closed affine subset; isomorphism between a principal open subset in A^{n}and a closed subset of A^{n+1}. To be closed in A^{n}is not invariant under isomorphisms. Affine varieties, examples of affine and non-affine varieties. If X and Y are affine and g:k[Y]->k[X] is a homomorphism of k-algebras, then there exists a unique morphism f:X->Y such that g=f^{*}. Application: affine varieties are determined by their coordinate rings. Affine open subsets are a basis for the topology of a quasi-projective variety. Example: the Veronese surface.Exercises: (1) the rational normal curve of degree d; (2) show that V(x

^{2}+y^{2}-1) in the affine plane is isomorphic to A^{1}minus a point (look at the projective closure); (3) S. ex. 5 and 11 p. 66-67 (end of Ch. I sec. 5).**Next Monday exercise session**### 2 April - 8 April

**3/4, 12.30-14.30:**Discussion of the exercises assigned in the previous lectures.**4/4, 12.30-14.30:**Products of quasi-projective varieties. The case of P^{1}xP^{1}. Segre maps and Segre varieties in general. A subset of P^{n}xP^{m}is closed if and only if it is the zero locus of some bihomogeneous polynomials. The Zariski topology on P^{n}xP^{m}is finer then the product topology. The product of two affine charts of P^{n}xP^{m }is isomorphic to A^{n+m}. The projections are regular, a product of two regular maps is regular. A product of affine varieties is affine. The diagonal is closed. The intersection of two affine open subsets is an affine open subset. The graph of a regular map is closed. Example: the Veronese map of degree 2.Exercises: (1) Show that a twisted cubic cannot be contained in a plane. (2) Show that a subset of A

^{n}xP^{m}is closed if and only if it is the zero locus of some polynomials in x_{1},...,x_{n},y_{0},...,y_{m}, homogeneous in the y's. (3) Show that the image of P^{n}x{pt} is a linear subspace of dimension n contained in the Segre variety. (4) Show that the irreducible components of a product are given by the products of the irreducible components of the two factors.**7/4, 10.30-12.30:**Rational functions on quasi-projective varieties: definition, examples, basic properties. If X is irreducible, the set k(X) of rational functions on X is a field. If X is irreducible and affine, then k(X) is isomorphic to the quotient field of k[X]. Examples: rational functions on A^{n}and on P^{n}. Rational maps between irreducible quasi-projective varieties. Examples. The Cremona map P^{2}-->P^{2}. Every rational map P^{1}-->P^{n}is regular. Dominant rational maps. Criterium for the definition of the composition of two rational maps. If either g is regular, or f is dominant, then g.f is defined. Birational maps, birational equivalence. Example: the Cremona map is birational.Exercises: (1) Show that the composition of dominant rational maps is dominant. (2) Show that birational equivalence is an equivalence relation on the set of irreducible q.p. varieties. (3) S. ex. 6 and 7 p. 40 (Ch. I sec. 3)

### 9 April - 15 April

**11/4, 12.30-14.30:**A rational map is birational if and only if it induces an isomorphisms between two non-empty open subsets; X and Y are birational if and only if they contain isomorphic non-empty open subsets. Pull-back of rational functions under a dominant rational map. If X and Y are irreducible, and g:k(Y)->k(X) is a k-linear field homomorphism, there exists a unique dominant rational map f:X-->Y such that g=f^{*}. A dominant rational map f is birational if and only if f^{*}is an isomorphism; X and Y are birational if and only if k(X) and k(Y) are isomorphic over k. Rational varieties; examples: the nodal and cuspidal cubic curves; P^{n}xP^{m}. Projection from a point and from a linear subspace in the projective space. Cones with vertex a point or a linear space. A quadric Q in P^{n}of rank r+1 is the cone over a quadric in P^{r}with vertex Sing(Q); examples in low dimensions. Interpretation of the isomorphism of an irreducible conic C with P^{1}as the projection from a point of C.Exercise: give an interpretation of the rational maps C

_{1}-->A^{1}and C_{2}-->A^{1}, where C_{1}and C_{2}are the nodal and cuspidal plane cubics, as projections from a point P^{2}-->P^{1}.### 16 April - 22 April

**21/4, 14.30-16.30:**Birational map from P^{1}xP^{1 }to P^{2}given by the projection from a point. If X is a closed subset of the projective space, and f:X->Y is a morphism, then f(X) is closed in Y. Applications: to be closed in the projective space is invariant under isomorphism; every regular function on a connected projective variety is constant. If X is a closed subset of the projective space, the projection XxY->Y is a closed map. Proof (first part).Exercises: (1) Show that every irreducible quadric is rational, using the projection from a point of the quadric. (2) Show that if X is both affine and projective, then X is finite. (3) Show that A

^{n}xP^{m}is neither affine nor projective (m,n positive).### 23 April - 29 April

**28/4, 14.30-16.30:**End of the proof started in the previous lecture. The Veronese map. Finitely generated field extensions, algebraic dependence and independence, trascendence bases. If L is finitely generated over k, then every trascendence basis is finite and has the same order (without proof); trascendence degree. Dimension of a quasi-projective variety. Examples. The dimension is a birational invariant. A hypersurface in A^{n }or P^{n}has pure dimension n-1. A variety X has dimension zero if and only if it is finite. If X is irreducible, and Y is a proper closed subset, then dim(Y)<dim(X).Exercise: let f:P

^{2}-> P^{5 }be the Veronese map, and L a line in P^{2}. Show that f(L) is a plane conic in P^{5}.### 30 April - 6 May

**2/5, 12.30-14.30:**A closed subset of A^{n}or P^{n}of pure dimension n-1 is a hypersurface. If there exists a dominant rational map X-->Y, then dim(X) is at least dim(Y). The dimension of XxY is dim(X)+dim(Y). Every irreducible variety is birational to a hypersurface (using the primitive element theorem with no proof).

Exterior products and exterior algebra of a vector space.**Exercise:**S. n. 11 p. 54 (Ch. I sec. 4).**5/5, 14.30-16.30:**Grassmannians, the Plucker map, the image of the Plucker map is a closed subset. Example: the grassmannian of lines in P^{3}, equation in the Plucker coordinates. Special loci: lines through a fixed point, lines contained in a plane.**Exercise:**let l be the line x_{2}=x_{3}=0 in P^{3}, and S the locus of lines in G(1,3) intersecting l. Determine S in P^{5}.### 7 May - 13 May

**No lectures this week**### 14 May - 20 May

**16/5, 12.30-14.30:**The grassmannian G(r,n) is an irreducible, rational variety, of dimension (r+1)(n-r). Finite and integral extensions of rings. If B is a finitely generated k-algebra, and A is a sub-algebra, then B is finite over A iff B is integral over A; moreover integrality can be checked for a set of generators of B (as a k-algebra) (no proof). Finite morphisms between affine varieties. If f is finite, then f is surjective, closed, and has finite fibers. Example. If f is a finite morphism between irreducible affine varieties, then f induces an algebraic field extension between the fields of rational functions, and dim(X)=dim(Y). The converse does not hold; example. If f:X->Y is finite, and U is a principal open subset of Y, then the inverse image V of U is a principal open subset of X, and the restriction of f V->U is still finite (proof in the irreducible case).**Exercises:**(1) Let f:A^{1}->C=V(y^{2}-x^{3}) be given by f(t)=(t^{2},t^{3}). Is f finite? (2) Let f:X-->Y be a dominant rational map between irreducible varieties. Show that dim(X)=dim(Y) if and only if the field extension induces by f is algebraic.**19/5, 14.30-16.30:**Discussion of the exercises assigned in the previous lectures.

If f:X->Y is a morphism between affine varieties such that Y has an affine open cover given by subsets U such that the inverse image V of U is affine and the restriction of f V->U is finite, then f is finite (proof in the irreducible case).### 21 May - 27 May

**22/5, 12.30-14.30:**Finite morphisms between quasi-projective varieties. If X is a closed subset in the projective space, and L is a linear subspace which is disjoint from X, then the projection from L is finite onto its image. If X is a closed subset in the projective space, and F_{0},...F_{r}are homogeneous polynomial of the same degree with no common zeros on X, then they define a morphism X->P^{r}which is finite onto its image. Noether's normalization lemma. Dimension of the intersection of a projective closed subset with a hypersurface. Applications: existence of subvarieties of any dimension between 0 and dim(X); topological definition of dimension.**23/5, 12.30-14.30:**Dimension of a projective closed subset X in terms of the maximal dimension of a linear subspace disjoint from X. Other applications of the result on the dimension of the intersection of a projective closed subset with a hypersurface. P^{2}and P^{1}xP^{1}are not isomorphic. Refined statement on the dimension of the irreducible components of the intersection of a projective closed subset with a hypersurface (no proof). Quasi-projective case (no proof). Intersection of two projective closed subsets.

Dimension of the fibers of a surjective morphism f:X->Y between irreducible varieties: if r is the difference of the dimensions of X and Y, every irreducible component of every fiber has dimension at least r, and there is a non-emtpy open subset of Y such that every fiber of f every U has dimension r.### 28 May - 3 June

**29/5, 12.30-14.30:**Upper semicontinuity of fiber dimension (no proof). Irreducibility criterion for the domain of a morphism.

In the projective space parametrizing surfaces of degree d in P^{3}, the locus parametrizing irreducible surfaces is a (non-empty) open subset. Lines on surfaces in P^{3}via the study of the incidence diagram. For d>3, a general surface of degree d in P^{3}does not contain lines.Exercise: for every n>2, find d(n) such that for every d>d(n), the general hypersurface of degree d in P

^{n}does not contain lines.**30/5, 12.30-14.30:**Action of projective transformations of P^{3}on the grassmannian of lines. Lines on planes and quadrics in P^{3}. Example of a cubic surface containing 3 lines. Every cubic surface contains lines, and the general one contains finitely many lines. A hypersurface X in P^{n}, which contains a linear subspace of dimension at least n/2, is singular.

Tangent space of an affine closed subset. The local ring of a quasi-projective variety at a point. If X is affine, every element in the local ring can be represented as a quotient of regular functions on X. The differential of a regular function as a linear form on the tangent space, for an affine closed subset.Exercises: S. n. 13 p. 82 (Ch. I, section 6), n.4,5,6 p. 96 (Ch. II, section 1).

### 4 June - 10 June

**6/6, 12.30-14.30:**If X is a closed affine subset and p a point in X, the differential induces an isomorphism of vector spaces between m/m^{2}and the dual of the tangent space. Isomorphism between the tangent space and the dual of m/m^{2}. Definition of tangent space for an arbitrary quasi-projective variety. Differential of a morphism. Relation between the tangent space of X at p, and the tangent space of an open/closed subset of X containing p. Upper semicontinuity of the dimension of the tangent space. When X is irreducible, there is a non-empty open subset where the dimension of the tangent space is equal to the dimension of X. Local dimension of a variety at a point. The dimension of the tangent space is always bigger or equal to the local dimension. Definition of singular and non-singular point. Examples. If X is irreducible, the singular locus is a proper closed subset.