## 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.

Pagina campusnet

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 An, 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 TpX. 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=X1u...uXr irreducible components, then Sing(X) is a closed subset given by the union of Sing(Xi) with the intersections among any two distinct components. If X is a complex algebraic hypersurface in An, 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 An, 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 An+m is (strictly) finer than the product of the Zariski topologies in An and Am. 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 An. Show that the intersection of X and Y is isomorphic to the intersection of XxY with the diagonal, in A2n.

• ### 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 Pn 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 P3.

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,t2,t3)} in A3 and let X be the projective closure of Y in P3. 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 Pn 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 P5 parametrizing conics of rank <3; closed locus Z in P5 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 PN. (3) Show that given a quadric Q, if f is the symmetric bilinear form on kn+1 associated to Q, then Sing(Q) is the projective subspace corresponding to ker(f). (4) Show that Z=Sing(D) in P5. (5) Let C be a conic given by the union of two distinct lines L1 and L2 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 Pn. If (n,d) is different from (2,2), give an example of an irreducible singular hypersurface of degree d in Pn. 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 Pn 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 An and a closed subset of An+1. To be closed in An 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(x2+y2-1) in the affine plane is isomorphic to A1 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 P1xP1. Segre maps and Segre varieties in general. A subset of PnxPm is closed if and only if it is the zero locus of some bihomogeneous polynomials. The Zariski topology on PnxPm is finer then the product topology. The product of two affine charts of PnxPm is isomorphic to An+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 AnxPm is closed if and only if it is the zero locus of some polynomials in x1,...,xn,y0,...,ym, homogeneous in the y's. (3) Show that the image of Pnx{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 An and on Pn. Rational maps between irreducible quasi-projective varieties. Examples. The Cremona map P2-->P2. Every rational map P1-->Pn 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; PnxPm. 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 Pn of rank r+1 is the cone over a quadric in Pr with vertex Sing(Q); examples in low dimensions. Interpretation of the isomorphism of an irreducible conic C with P1 as the projection from a point of C.

Exercise: give an interpretation of the rational maps C1-->A1 and C2-->A1, where C1 and C2 are the nodal and cuspidal plane cubics, as projections from a point P2-->P1.

• ### 16 April - 22 April

21/4, 14.30-16.30: Birational map from P1xPto P2given 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 AnxPm 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 Aor Pn 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:P2 -> P5 be the Veronese map, and L a line in P2. Show that f(L) is a plane conic in P5.

• ### 30 April - 6 May

2/5, 12.30-14.30: A closed subset of An or Pn 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 P3, equation in the Plucker coordinates. Special loci: lines through a fixed point, lines contained in a plane.

Exercise: let l be the line x2=x3=0 in P3, and S the locus of lines in G(1,3) intersecting l. Determine S in P5.

• ### 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:A1->C=V(y2-x3) be given by f(t)=(t2,t3). 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 F0,...Fr are homogeneous polynomial of the same degree with no common zeros on X, then they define a morphism X->Pr 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. P2 and P1xP1 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 P3, the locus parametrizing irreducible surfaces is a (non-empty) open subset. Lines on surfaces in P3 via the study of the incidence diagram. For d>3, a general surface of degree d in P3 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 Pn does not contain lines.

30/5, 12.30-14.30: Action of projective transformations of P3 on the grassmannian of lines. Lines on planes and quadrics in P3. 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 Pn, 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/m2 and the dual of the tangent space. Isomorphism between the tangent space and the dual of m/m2. 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.