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  • Geometria Algebrica

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

    Docente: Cinzia Casagrande

    Orario: martedi' 12:30-14:30, venerdi' 10:30-12:30, aula 2

    Important notices on the schedule:
    on  Fri 27/4, Fri 4/5, Fri 11/5, and 
    Fri 18/5 the course will not take place
    instead there will be lectures on: Mon 16/4, Mon 7/5, Mon 28/5 at 16:30-18:30 (room 2)

    Lunedi' 21/5, h 12:30 - 13:30, aula A: Seminario Math-Lab di Geometria Algebrica "Tori complessi, varieta' abeliane  e varieta' proiettive irregolari", Prof. Rita Pardini (Universita' di Pisa) (vedi file qui sotto)

    Ricevimento studentisu appuntamento (da concordare a voce o per email).

    Pagina campusnet

    Programma e testi consigliati:
    Il corso si propone di coprire il capitolo I del libro Algebraic Geometry di Hartshorne.
    Altri testi di riferimento (sugli stessi argomenti) sono:
    Shafarevich, Basic Algebraic Geometry, vol. 1
    Reid, Undergraduate Algebraic Geometry

    Referenze di algebra, ove necessarie:
    Lang, Algebra
    Atiyah e Macdonald, Introduction to Commutative Algebra 
    Reid, Undergraduate Commutative Algebra (evidenzia le relazioni con la geometria algebrica)
  • 26 febbraio - 4 marzo

    27/2, 12:30-14:30  Introduzione al corso.
    Richiami dal corso di Istituzioni di Geometria: chiusi algebrici affini, topologia di Zariski sullo spazio affine. Nullstellensatz. Proprieta' principali delle mappe V e I.
    Spazio topologico irriducibile, caratterizzazioni equivalenti. Un chiuso affine e' irriducibile sse il suo ideale e' primo. Spazi topologici noetheriani. Uno spazio topologico noetheriano e' compatto. Decomposizione in componenti irriducibili di uno spazio topologico noetheriano.
    Esercizi: 1) Dato un sottoinsieme Y denso in uno spazio topologico X, Y e' irriducibile sse lo e' X. 2) Ogni sottoinsieme di uno spazio topologico noetheriano e' noetheriano (con la topologia indotta). 3) Hartshorne, es. I.1.3.

    2/3, 10:30-12:30  The irreducible components are the maximal irreducible closed subsets of a noetherian topological space.
    Hartshorne, ex. I.1.3.
    k-algebras, finitely generated k-algebras. The coordinate ring k[X] of a closed affine subset X. The coordinate ring is a f.g. k-algebra, without nilpotents. Conversely, every f.g. k-algebra without nilpotents is isomorphic to the coordinate ring of some closed affine subset. Bijection between closed subsets of X and radical ideals of k[X].
    Dimension of a topological space. A point has dimension zero, the affine line has dimension one. Example of a noetherian topological space with infinite dimension. Properties of topological dimension: 1) for every subset Y of X, dim(Y) is at most dim (X); 2) if X is noetherian, dim(X) is the maximum of the dimensions of its irreducible components; 3) if X is irreducible of finite dimension, and Y is a proper closed subset, then dim(Y)<dim(X).
    Krull dimension of a ring. The dimension of a closed affine subset is equal to the Krull dimension of its coordinate ring. 
    Exercises: 1) Let X be a topological space. Show that X is noetherian and Hausdorff iff X is finite with the discrete topology. 2) Let f,g two polynomials in k[An] such that V(f)=V(g). What can be said about f and g? 3) Shafarevich, Ch. I section 3 ex. 1.

    • 5 marzo - 11 marzo

      6/3, 12:30-14:30  Finitely generated field extensions. Algebraic dependence and independence. Trascendence bases, trascendence degree of a finitely generated field extension (no proof). Given a f.g. k-algebra A which is a domain, if K is its quotient field, then dim(A) is equal to the trascendence degree of K over k (no proof). Applications: dim(An)=n; every closed affine subset (and every open subset of a closed affine subset) has finite dimension.
      Exercise I.1.1 from Hartshorne.
      Height of a prime ideal. Given a f.g. k-algebra A which is a domain, and a prime ideal p of A, then ht(p)+dim(A/p)=dim(A) (no proof). Geometric consequence: given a closed affine subset X of dimension n, and an irreducible closed subset Z of X, there exists a chain of irreducible closed subsets of X, of length n+1, containing Z.

      Homework: 1) Let X be an affine closed subset. Show that dim(X)=0 iff X is finite. 2) Hartshorne, ex. I.1.2 (twisted cubic). 3) Let X be an irreducible curve (closed in An, or open in its closure in An). Show that X has the cofinite topology.

      9/3, 10:30-12:30  If Y is open in its closure Z in An, then dim(Y)=dim(Z). 
      Affine hypersurfaces. A prime ideal in the polynomial ring has height 1 iff it is principal and non-zero. An irreducible closed subset in An has dimension n-1 iff it is a hypersurface. Krull's principal ideal theorem (no proof). If A if a noetherian domain, A is a U.F.D. iff every prime ideal with height 1 is principal (no proof). Dimension of the intersection of an irreducible closed subset with a hypersurface. If a closed subset X of An is defined by r equations, then every irreducible component of X has dimension at least n-r. Example of an irreducible affine curve in A3 whose ideal cannot be generated by two elements.
      Grading on the polynomial ring. Homogeneous ideals; an ideal is homogeneous iff it can be generated by homogeneous polynomials.

      Homework: Identify A2 with A1xA1 as sets. Show that the Zariski topology in A2 is strictly finer than the product of the Zariski topologies.

      Next lecture: discussion of homework exercises

      • 12 marzo - 18 marzo

        13/3, 12:30-14:30  Discussion of homework exercises.
        Zariski topology on the projective space. Projective varieties and quasi-projective varieties. Homogenization and dehomogenization of polynomials.

        Homework: 1) Given a polynomial f, show that the radical of the ideal generated by f is principal, generated by the product of the irreducible factors of f. 2) Given a homogeneous polynomial F, show that every irreducible factor of F is homogeneous. 3) Show that sum, intersection, and radical of homogeneous ideals are homogeneous. 4) Let I be a homogeneous ideal. Show that I is prime iff for every F,G homogeneous polynomials such that FG is in I, then at least one of F and G is in I. 5) Show that the Zariski topology on the projective space is a topology. 6) Let X,Y be quasi-projective varieties in Pn. Show that the intersection of X and Y is a q.p. variety, while the union of X and Y does not need to be a q.p. variety.

        16/3, 10:30-12:30  Homeomorphism between the affine chart on Pn and An.
        Affine cone over a closed projective subset. Projective Nullstellensatz: V(I) is empty iff the radical of I contains the irrelevant ideal. Ideal of a subset of Pn. Properties of the function V and I in the projective case. Bijection between closed subsets of Pn and radical, homogeneous ideals of S (irrelevant ideal excluded).
        The projective space is irreducible and noetherian. Every q.p. variety is noetherian and has a decomposition in irreducible components.
        Given a topological space X and an open cover {Ui}, the dimension of X is Sup dim(Ui).
        Pn has dimension n and every q.p. variety has finite dimension.
        Homogeneous coordinate ring S(X) of a closed projective subset X; dim X=dim S(X)-1.
        Projective closure of a closed affine subset. Example: the twisted cubic in P3.

        Homework: Let X be a projective closed subset. Show that X is irreducible iff I(X) is prime.

        • 19 marzo - 25 marzo

          20/3, 12:30-14:30  If X is a q.p. variety, dim(X) is equal to the dimension of the closure of X.
          Twisted cubic X in P3: description as the image of a map from P1, explicit computation of I(X).
          Examples of projective closed subsets: linear spaces, hypersurfaces. Projectively equivalent closed subsets. Classification of quadrics by the rank, up to projective transformation (char k not 2). An irreducible plane conic as the image of a map from P1.

          Homework: (1) Let X be an irreducible q.p. variety and U a non-empty open subset. Show that dim(U)=dim(X). (2) Let X={(t2,t3,t4)|t in k}. Show that X is closed in A3 and determine I(X). (3) Let Y={(t3,t4,t5)|t in k} in A3. Determine I(Y). (4) Let Q be a quadric of rank at least 3. Show that Q is irreducible. (5) Rational normal curves of degree d: given the map f:P1 -> Pd, set X=Im f. Show that X is closed, that f is a homeomorphism, that X is an irreducible curve, and that X is not contained in a hyperplane.

          23/3, 10:30-12:30  A closed subset of Pn is a hypersurface iff it has pure dimension n-1.
          Examples: the ideal of one or two points in P2
          The ideal of a union of closed subsets in general is not the sum of the ideals of the closed subsets.
          Projective cones; example: quadrics with non-maximal rank. 
          The Veronese surface in P5.
          If a non-empty closed subset X in Pn is defined by r equations, then every irreducible component of X has dimension at least n-r. Complete intersections and set-theoretic complete intersections. Example: the twisted cubic in P3 is not a complete intersection, but it is a set-theoretic complete intersection.

          Homework: (1) Show that (xy,z) is the intersection of the ideals (x,z) and (y,z). (2) Let X={three distinct points} in P2. Determine I(X) (up to projective equivalence). (3) Let v:P2->S be the map onto the Veronese surface. Show that for every line l in P2 there exists a plane P(l) in P5 such that v(l) is the intersection of S with P(l), and v(l) is a conic in P(l). (4) Show that for every irreducible conic C in P2 there exists a hyperplane H(C) in P5 such that v(C) is the intersection of S with H(C), and v(C) is a rational normal curve of degree 4 in H(C). (Hint for (3) and (4): choose suitable projective coordinates in P2.)  Let C be the twisted cubic in P3. Show that C=V(y2-xz, z(z2-yw)+w(xw-yz) ).

          • 26 marzo - 1 aprile

            27/3, 12:30-14:30  The Veronese map v of degree d: the image is closed, v is injective.
            The Segre map and the Segre variety. The Segre quadric in P3, the families of lines.

            Homework: (1) Show that the Veronese map v is a homeomorphism onto its image. (2) Let S be the Segre quadric in P3. Find a curve in S different from the lines of the two families, and hence show that the topology on S is not homeomorphic to the product topology.

            • 2 aprile - 8 aprile

              6/4, 10:30-12:30:  Discussion of homework exercises.
              The parameter space for quadrics in P3; quadrics containing a given line. Exercise: given three lines in P3, there exists a quadric containing them.
              The Grassmannian of lines in Pn: Plucker map, Plucker coordinates, equations. The Grassmannian of lines in P3.

              Homework: Let G be the Grassmannian of lines in P3. Describe the following subsets, and show that they are closed:
              (1) lines contining (0:0:0:1)
              (2) lines contained in V(x3)
              (3) lines meeting V(x2,x3)

              • 9 aprile - 15 aprile

                10/4, 12:30-14:30  Regular functions on locally closed subsets of An, as functions which are given locally by quotients of polynomials. Regular functions on quasi-projective varieties, as functions which are given locally as quotients of homogeneous polynomials of the same degree. The two definitions coincide when we identify An with an affine chart of Pn. Regular function are continuous. The set O(X) of regular functions on X is a k-algebra. If X is irreducible and two regular functions f and g coincide on a non-empty open subset, then f=g.
                Morphisms between quasi-projective varieties, as a continuous maps which preserves regular functions. Isomorphisms.
                Germs of regular functions, local ring Op of an irreducible variety at a point p.
                Field of rational functions k(X) of an irreducible variety X.
                Natural k-algebra homomorphisms O(X) -> Op -> k(X).
                Pull-backs defined by a morphism; dominant morphism. Functoriality: the ring of regular functions, the local ring at a point, and the field of rational functions, are invariant under isomorphisms.
                Localization AP of a domain A at a prime ideal P. The localization is a local ring. A is equal to the intersection (in its fraction field) of the localizations at all maximal ideals. Bijection between ideals in AP and ideals in A contained in P (no proof). The dimension of AP is equal to the height of P.
                If X is a closed subset of the affine space, then O(X) is isomorphic to the coordinate ring k[X], (X irreducible) Op is isomorphic to the localization of k[X] at the maximal ideal corresponding to p, and k(X) is isomorphic to the fraction field of k[X]. Moreover the dimension of Op is equal to dim(X).

                13/4, 10:30-12:30  If Y is contained in Am, a map f:X->Y is a morphism iff the components of f are regular functions on X. Regular functions on A1\{0}. To be closed in the affine space is not invariant under isomorphism. Definition of affine variety as a q.p. variety which is isomorphic to a closed subset of an affine space. Properties of the ring of regular functions of an affine variety. If X is q.p. and Y is affine, then there is a bijection between Mor(X,Y) and Hom(O(Y),O(X)). Applications: (1) if f:X->Y is a morphism between affine varieties, then f is an isomorphism iff f*:k[Y]->k[X] is an isomorphism; (2) given X,Y affine varieties, X is isomorphic to Y iff k[X] is isomorphic to k[Y]; (3) there is an equivalence of categories between affine varieties and f.g. k-algebras without nilpotents.

                • 16 aprile - 22 aprile

                  16/4, 16:30 - 18:30 (room 2) An irreducible conic in A2 is isomorphic to A1 or to A1\{0}. The morphism from A1 to the cuspidal plane cubic. The Frobenius morphism A1->A1. Automorphisms of An and the Jacobian conjecture.
                  Characterization of morphisms as maps which are given locally by homogeneous polynomials of the same degree.
                  An irreducible plane conic in P2 is isomorphic to P1. The homoheneous coordinate ring is not invariant under isomorphism.
                  The complement of a hypersurface in An is an affine variety (principal open subsets of An).

                  Homework: (1) Let C=V(y2-x2-x3) in A2 and f:A1->C given by f(t)=(t2-1,t(t2-1)). Show that f is a surjective morphism and describe its fibers. (2) Given a map f:X->Y and an open cover {Ai} of X, if the restriction of f Ai->Y is a morphism for every i, then f is a morphism. (3) Show that the map f:P1->C, where C is the rational normal curve, is an isomorphism. (4) Show that the map f:P2->S, where S is the Veronese surface in P5, is an isomorphism. (5) Hartshorne ex. I.3.5: the complement of a hypersurface in Pn is an affine variety.

                  17/4, 12:30 - 14:30  Let X be a closed projective subset, and f:X->Y be a morphism, with Y quasi projective. Then f is a closed map (without proof).
                  Applications: to be closed in Pn is invariant under isomorphism; on a connected projective variety, every regular function is constant; a variety which is both projective and affine is finite. Example of a variety which is neither projective nor affine: A2\{(0,0)}. Two plane projective curves must always intersect. Projection from a point in Pn. Projection of the twisted cubic onto the cuspidal plane cubic.
                  Affine open subsets. For every q.p. variety, affine open subsets form a basis for the topology.
                  Products of closed affine subsets. Topology on PnxPm induced by the Segre map. Bihomoheneous polynomials. Closed subsets of PnxPm are given by zero loci of bihomogeneous polynomials.

                  Homework: (1) Let f:X->Y be a dominat map of quasi-projective varieties. Show that f*:O(Y)->O(X) is injective, and that if X is irreducible, then Y is irreducible. (2) Show that the restriction k[A2]->O(A2\{(0,0)}) is an isomorphism.

                  20/4, 10:30 - 12:30  The topology of PnxPm contains the product topology. Products of quasi-projective varieties. The Segre map on the affine charts is an isomorphism. The projections are morphisms. A product XxY is irreducible iff X and Y are irreducible. Tensor product of k-algebras; if X and Y are affine, k[XxY] is isomorphic to the tensor product of k[X] and k[Y] (no proof). The diagonal in XxX is closed. If X is irreducible and two morphism X->Y coincide on a non-empty open subset, then they are equal.
                  Rational maps between irreducible varieties; dominant rational maps; domain of a rational map. Given rational maps f:X-->Y and g:Y-->Z, if g is regular or f is dominant, then the composition is a well-defined rational map X-->Z. Birational maps, birational equivalence. Examples: the Cremona map on P2; the map from A1 to the cuspidal cubic; the inclusion of a non-empty open subset in X.

                  Homework: (1) Show that Xx{y} in XxY is isomorphic to X. (2) Show that the irreducible components of XxY are the products of the irreducible components of X and Y. (3) Show that the intersection of two affine open subsets in a q.p. projective variety X is affine. (4) Show that A1xP1 is neither affine nor projective. 


                  • 23 aprile - 29 aprile

                    24/4, 12:30 - 14:30: A dominant rational map induces a k-algebra homomorphism between the function fields. Example: the field of rational functions of the projective space. Every f.g. field extension of k is the field of rational function of an irreducible q.p. variety. Given X and Y irreducible q.p. varieties, there is a bijection between the set of dominant rational maps X-->Y and the set of k-algebra homomorphisms k(Y)->k(X). Applications to birational maps and birational equivalence. Two varieties are birationally equivalent iff they have two non-empty isomorphic open subsets.
                    The primitive element theorem (no proof). Application: every irreducible variety is birational to a hypersurface.
                    Construction of the blow-up of An at the origin.

                    Homework:  Show that a subset Z of AnxPm is closed iff Z is the zero locus of F1,...,Fr, where each Fi is a polynomial in the coordinates of An and Pm, homogeneous in the coordinates of Pm.

                    No lecture on Friday 27/4

                    On Monday 7/5: discussion of homework exercises


                    • 30 aprile - 6 maggio

                      No lectures this week

                      • 7 maggio - 13 maggio

                        7/5, 16:30 - 18:30  Discussion of homework exercises.
                        The blow-up of An at the origin. Proper transform. The proper transform of the nodal cubic in the blow-up of A2 at the origin.
                        Rational varieties. Examples of rational curves: conics, singular cubics. P1xP1 is rational but not isomorphic to P2.
                        Singular points of an irreducible closed subset of AN, via the rank of the jacobian matrix. Examples.

                        Homework: (1) Show that the blow-up of A2 at the origin is neither affine nor projective. (2) Let C be the cuspidal cubic in A2; determine the proper transform of C in the blow-up of A2 at the origin, and its intersection with the exceptional curve.

                        8/5, 12:30 - 14:30  Noetherian local rings A: structure of vector space on m/m2. The dimension of m/m2 is finite and is at least dim(A) (no proof). Regular local rings. If X is an irreducible closed subset of AN, and p a point of X, then X is non singular at p iff OX,p is a regular local ring (partial proof). If X is an irreducible variety, Sing(X) is a proper closed subset. A point p in a q.p. variety is non-singular iff it belongs to a unique irreducible component X0, and X0 is smooth at p. Local dimension, characterization of singular points of a closed affine subset in terms of the jacobian (no proof), examples.

                        No lecture on Friday 11/5

                        • 14 maggio - 20 maggio

                          15/5, 12:30 - 14:30  If X is a closed irreducible subset of AN of dimension n, and p is a non-singular point of X, then there exists an open neighborhood U of p in AN and polynomials F1,...,FN-n such that X in U is the zero locus of F1,...,FN-n, and the jacobian matrix has rank N-n at p.
                          Topological properties of complex q.p. varieties with respect to the euclidean topology. An irreducible complex q.p. variety is connected for the euclidean topology (no proof). A complex q.p. variety is projective iff it is compact for the euclidean topology.
                          Smooth complex q.p. varieties versus complex manifolds: given X an irreducible smooth complex q.p. variety of dimension n, construction of an atlas on X as complex manifold of dimension n, via the implicit function theorem: associated complex manifold Xan. Discussion of GAGA in the projective case: if X is a smooth, irreducible, complex projective variety, every global holomorphic function is constant, every global meromorphic function is rational, and X and Y are isomorphic iff Xan and Yan are biholomorphic (no proofs). Every compact Riemann surface is biholomorphic to Xan for some smooth complex projective curve X, while in higher dimension compact complex manifolds are a larger class than smooth complex projective varieties (no proofs).
                          Definition of algebraic group; examples: An, GL(n,k). Every algebraic group is smooth. Abelian varieties: every projective, irreducible algebraic group is abelian (no proof). If X is a complex abelian variety, then Xan is a complex torus (no proof). Discussion of complex tori and complex abelian varieties in dimension 1 and dimension >1.

                          Homework: Hartshorne ex. I.4.3, I.4.7.

                          No lecture on Friday 18/5

                          • 21 maggio - 27 maggio

                            Lunedi' 21/5, h 12:30, aula A: Seminario Math-Lab di Geometria Algebrica "Tori complessi, varieta' abeliane  e varieta' proiettive irregolari", Prof. Rita Pardini (Universita' di Pisa)

                            22/5, 12:30 - 14:30  Informal discussion on the projectivity condition for complex tori, following Rita Pardini's talk. 
                            Back to algebraic groups: every affine algebraic group is a linear group, that is: a closed subgroup of some general linear group (no proof).
                            Topological genus of a smooth complex projective curve; discussion of the cases g=0 and g=1 with no proofs. Relation among 1-dimension complex tori / elliptic curves / genus 1 curves. Genus of a plane curve in terms of the degree.
                            Zariski tangent/cotangent space of an irreducible q.p. variety at a point. Embedded tangent space for a closed subset of AN at a point. Differentials of regular functions on X as linear forms on the embedded tangent space; extension to the local ring. The differential induces an isomorphism of vector spaces between the Zariski cotangent space and the dual of the embedded tangent space. Induced isomorphism between the Zariski tangent space and the embedded tangent space; explicit description. Differential of a morphism; explicit description in terms of the jacobian for closed subsets of the affine space.

                            Homework: (1) Explicit description of the differential of a morphism between closed subsets of the affine space, in terms of the jacobian. (2) Let X=V(y2-x3) in A2, and f:A1->X given by f(t)=(t2,t3). Describe the differential of f at p, for every point p in A1.  (3) Let Q=V(x0x3-x1x2) in P3, G the Grassmannian of lines in P3, and X the subset of G given by lines contained in Q. Show that X is closed in G, and that X is the disjoint union of two conics.

                            Reference for tangent spaces: Shafarevich, Ch. II, section 1

                            25/5, 10:30 - 12:30  Exercise 2 from last lecture homework. Projective tangent space, case of hypersurfaces. Singular locus of a projective hypersurface. If F is a homogeneous polymial, with partial derivatives F0,...,Fn, if V(F, F0,...,Fn) has dimension at most n-3 in Pn, then F is irreducible. In the projectivization PN of the vector space of degree d homogeneous polynomials, the loci M of non-reduced polynomials and R of reducible polynomials are closed subsets. Parameter space for degree d projective hypersurfaces; the general hypersurface is irreducible.
                            Given r homogeneous polynomials in k[x0,...,xn] with r<n+1, their common zero locus X in Pn is non-empty, and every irreducible component of X has dimension at least n-r.
                            A reducible hypersurface X is always singular, unless n=1 and X is finite.
                            Subset S of PN given by non-reduced polynomials and reduced polynomials F such that V(F) is singular. Proof that S is closed, via the incidence diagram in PnxPN (first part).

                            Homework: For every pair of integers n,d>1, give an example of a non-singular hypersurface of degree d in Pn.

                            • 28 maggio - 3 giugno

                              28/5, 16:30-18:30, room 2: lecture/seminar by Iman Bahmani on: The theorem on dimension of fibers and applications to lines on hypersurfaces

                              29/5, 12:30-14:30: If f:X->Y is a surjective morphism between irreducible varieties, then the general fiber has dimension dim(X)-dim(Y), and the dimension of the fiber is upper semicontinuous (no proof). If f:X->Y is a surjective morphism between projective varieties, such that Y is irreducible and every fiber of f is irreducible of dimension m, then X is irreducible and m=dim(X)-dim(Y) (no proof). 
                              The subset S of singular hypersurfaces of degree d in Pn is an irreducible hypersurface in PN. The general singular hypersurface has finitely many singular points. Example: the case of plane conics.
                              Lines on surfaces in P3: study via the incidence diagram. The general surface of degree d>3 does not contain lines. Study of the cases d=1, d=2, d=3.
                              If X is a non-singular irreducible variety and f:X-->PN a rational map, then the locus where f is not regular has codimension at least 2 (only statement). Sharpness of the statement. Applications: every rational map form a smooth curve to the projective space is regular; two irreducible smooth projective curves are isomorphic iff they are birationally equivalent.

                              Homework: (1) Consider the P5 parametrizing 3x3 symmetric matrices up to scalars (hence plane conics), and let X be the cubic hypersurface given by the determinant. Show that the singular locus of X is the locus of rank 1 matrices (Veronese surface). Let C=V(x0x1) and [C] in X. Show that the projective tangent space to X at [C] is the hyperplane of conics containing the point (0:0:1).
                              (2) (premio in palio per questo esercizio) Let X be a hypersurface of degree >1 in Pn, and suppose that X contains a linear subspace L with dim(L) at least n/2. Show that X is singular.

                              1/6, 10:30-12:30: The local ring at a non-singular point is a UFD (no proof). The ideal of a hypersurface in a smooth variety is locally generated by one equation (no proof). Sketch of proof that a rational map from a smooth variety to the projective space is regular in codimension one.
                              Overview of birational geometry over the complex numbers: resolution of singularities. Genus of a smooth projective curve, moduli spaces of genus g curves. Higher dimensions: blow-up of a point. Birational equivalence and birational morphisms. Factorization of the birational map from P1xP1 to P2, and of the Cremona map, as a sequence of blow-ups. Graph given by varieties and birational morphisms; second Betti number; minimal varieties as distinguished smooth projective varieties in a given birational equivalence class.

                              Homework: Let X be a quadric in the projective space. Show that X is non-singular iff X has maximal rank. More generally, show that the singular locus of X is a linear subspace, given by the projectivization of the kernel of the symmetric bilinear form associated to X, so that the codimension of Sing(X) is the rank of X.

                              • 4 giugno - 10 giugno

                                5/6, 12:30-14:30: Overview of birational geometry: intersection curves/hypersurfaces; real 1-cycles, vector space N1(X) of numerical equivalence classes of 1-cycles. Examples: projective plane, smooth quadric surface. Cone of effective 1-cycles. Face of the cone of effective 1-cycles associated to a morphism.
                                Canonical class of a smooth projective variety: meromorphic n-form, zeros and poles, linear form KX defined on N1(X). Example: the projective plane. Varieties where KX has a special behaviour with respect to the cone of effective curves. Varieties with KX nef, minimal varieties. If X is minimal, every birational morphism X->Y, with Y smooth and projective, is an isomorphism.