## Weekly outline

### Geometria Algebrica

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

Docente:

__Cinzia Casagrande__martedi' 12:30-14:30, venerdi' 10:30-12:30, aula 2**Orario:***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)**su appuntamento (da concordare a voce o per email)**Ricevimento studenti**:**.****Programma e testi consigliati:**Il corso si propone di coprire il capitolo I del libro*Algebraic Geometry*di

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 February - 4 March

**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[A^{n}] such that V(f)=V(g). What can be said about f and g? 3) Shafarevich, Ch. I section 3 ex. 1.### 5 March - 11 March

**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(A^{n})=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 A^{n}, or open in its closure in A^{n}). Show that X has the cofinite topology.**9/3, 10:30-12:30**If Y is open in its closure Z in A^{n}, 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 A^{n}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 A^{n}is defined by r equations, then every irreducible component of X has dimension at least n-r. Example of an irreducible affine curve in A^{3}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 A^{2}with A^{1}xA^{1}as sets. Show that the Zariski topology in A^{2}is strictly finer than the product of the Zariski topologies.**Next lecture: discussion of homework exercises**### 12 March - 18 March

**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 P^{n}. 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 P^{n}and A^{n}.

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 P^{n}. Properties of the function V and I in the projective case. Bijection between closed subsets of P^{n}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 {U_{i}}, the dimension of X is Sup dim(U_{i}).

P^{n}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 P^{3}.*Homework:*Let X be a projective closed subset. Show that X is irreducible iff I(X) is prime.### 19 March - 25 March

**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 P^{3}: description as the image of a map from P^{1}, 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 P^{1}.*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={(t^{2},t^{3},t^{4})|t in k}. Show that X is closed in A^{3}and determine I(X). (3) Let Y={(t^{3},t^{4},t^{5})|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:P^{1}-> P^{d}, 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 P^{n}is a hypersurface iff it has pure dimension n-1.

Examples: the ideal of one or two points in P^{2}.

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 P^{5}.

If a non-empty closed subset X in P^{n}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 P^{3}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 P^{2}. Determine I(X) (up to projective equivalence). (3) Let v:P^{2}->S be the map onto the Veronese surface. Show that for every line l in P^{2}there exists a plane P(l) in P^{5}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 P^{2}there exists a hyperplane H(C) in P^{5}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 P^{3}. Show that C=V(y^{2}-xz, z(z^{2}-yw)+w(xw-yz) ).### 26 March - 1 April

**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 P^{3}, 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 P^{3}. 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 April - 8 April

**6/4, 10:30-12:30:**Discussion of homework exercises.

The parameter space for quadrics in P^{3}; quadrics containing a given line. Exercise: given three lines in P^{3}, there exists a quadric containing them.

The Grassmannian of lines in P^{n}: Plucker map, Plucker coordinates, equations. The Grassmannian of lines in P^{3}.*Homework:*Let G be the Grassmannian of lines in P^{3}. Describe the following subsets, and show that they are closed:

(1) lines contining (0:0:0:1)

(2) lines contained in V(x_{3})

(3) lines meeting V(x_{2},x_{3})### 9 April - 15 April

**10/4, 12:30-14:30**Regular functions on locally closed subsets of A^{n}, 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 A^{n}with an affine chart of P^{n}. 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 O_{p}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) -> O_{p}-> 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 A_{P}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 A_{P}and ideals in A contained in P (no proof). The dimension of A_{P}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) O_{p}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 O_{p}is equal to dim(X).**13/4, 10:30-12:30**If Y is contained in A^{m}, a map f:X->Y is a morphism iff the components of f are regular functions on X. Regular functions on A^{1}\{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 April - 22 April

**16/4, 16:30 - 18:30 (room 2)**An irreducible conic in A^{2}is isomorphic to A^{1}or to A^{1}\{0}. The morphism from A^{1}to the cuspidal plane cubic. The Frobenius morphism A^{1}->A^{1}. Automorphisms of A^{n}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 P^{2}is isomorphic to P^{1}. The homoheneous coordinate ring is not invariant under isomorphism.

The complement of a hypersurface in A^{n}is an affine variety (principal open subsets of A^{n}).*Homework:*(1) Let C=V(y^{2}-x^{2}-x^{3}) in A^{2}and f:A^{1}->C given by f(t)=(t^{2}-1,t(t^{2}-1)). Show that f is a surjective morphism and describe its fibers. (2) Given a map f:X->Y and an open cover {A_{i}} of X, if the restriction of f A_{i}->Y is a morphism for every i, then f is a morphism. (3) Show that the map f:P^{1}->C, where C is the rational normal curve, is an isomorphism. (4) Show that the map f:P^{2}->S, where S is the Veronese surface in P^{5}, is an isomorphism. (5) Hartshorne ex. I.3.5: the complement of a hypersurface in P^{n}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 P^{n}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: A^{2}\{(0,0)}. Two plane projective curves must always intersect. Projection from a point in P^{n}. 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 P^{n}xP^{m}induced by the Segre map. Bihomoheneous polynomials. Closed subsets of P^{n}xP^{m}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[A^{2}]->O(A^{2}\{(0,0)}) is an isomorphism.**20/4, 10:30 - 12:30**The topology of P^{n}xP^{m }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 P^{2}; the map from A^{1}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 A^{1}xP^{1}is neither affine nor projective.### 23 April - 29 April

**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 A^{n}at the origin.*Homework:*Show that a subset Z of A^{n}xP^{m}is closed iff Z is the zero locus of F_{1},...,F_{r}, where each F_{i}is a polynomial in the coordinates of A^{n}and P^{m}, homogeneous in the coordinates of P^{m}.**No lecture on Friday 27/4****On Monday 7/5: discussion of homework exercises**### 30 April - 6 May

**No lectures this week**### 7 May - 13 May

**7/5, 16:30 - 18:30**Discussion of homework exercises.

The blow-up of A^{n}at the origin. Proper transform. The proper transform of the nodal cubic in the blow-up of A^{2 }at the origin.

Rational varieties. Examples of rational curves: conics, singular cubics. P^{1}xP^{1}is rational but not isomorphic to P^{2}.

Singular points of an irreducible closed subset of A^{N}, via the rank of the jacobian matrix. Examples.*Homework:*(1) Show that the blow-up of A^{2}at the origin is neither affine nor projective. (2) Let C be the cuspidal cubic in A^{2}; determine the proper transform of C in the blow-up of A^{2}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/m^{2}. The dimension of m/m^{2}is finite and is at least dim(A) (no proof). Regular local rings. If X is an irreducible closed subset of A^{N}, and p a point of X, then X is non singular at p iff O_{X,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 X_{0}, and X_{0}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 May - 20 May

**15/5, 12:30 - 14:30**If X is a closed irreducible subset of A^{N}of dimension n, and p is a non-singular point of X, then there exists an open neighborhood U of p in A^{N}and polynomials F_{1},...,F_{N-n}such that X in U is the zero locus of F_{1},...,F_{N-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 X^{an}. 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 X^{an}and Y^{an}are biholomorphic (no proofs). Every compact Riemann surface is biholomorphic to X^{an}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: A^{n}, 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 X^{an}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 May - 27 May

**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 A^{N}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(y^{2}-x^{3}) in A^{2}, and f:A^{1}->X given by f(t)=(t^{2},t^{3}). Describe the differential of f at p, for every point p in A^{1}. (3) Let Q=V(x_{0}x_{3}-x_{1}x_{2}) in P^{3}, G the Grassmannian of lines in P^{3}, 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 F_{0},...,F_{n}, if V(F, F_{0},...,F_{n}) has dimension at most n-3 in P^{n}, then F is irreducible. In the projectivization P^{N}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[x_{0},...,x_{n}] with r<n+1, their common zero locus X in P^{n}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 P^{N}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 P^{n}xP^{N}(first part).*Homework:*For every pair of integers n,d>1, give an example of a non-singular hypersurface of degree d in P^{n}.### 28 May - 3 June

**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 P^{n}is an irreducible hypersurface in P^{N}. The general singular hypersurface has finitely many singular points. Example: the case of plane conics.

Lines on surfaces in P^{3}: 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-->P^{N}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 P^{5}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(x_{0}x_{1}) 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 P^{n}, 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 P^{1}xP^{1}to P^{2}, 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 June - 10 June

**5/6, 12:30-14:30:**Overview of birational geometry: intersection curves/hypersurfaces; real 1-cycles, vector space N_{1}(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 K_{X}defined on N_{1}(X). Example: the projective plane. Varieties where K_{X}has a special behaviour with respect to the cone of effective curves. Varieties with K_{X}nef, minimal varieties. If X is minimal, every birational morphism X->Y, with Y smooth and projective, is an isomorphism.