Wednesdays, 12pm(noon)-2pm, 3rd floor Seminar Room
Lecture 4: Divisors. Cartier and Weil divisors. Principal divisors and linear equivalence. Picard Group. References: [6] Examples: Pic(P^1) Pic(x^2+y^2 = 1) Homework problems: (3) Point on a circle is not a principal divisor. Lecture 5: Invertible sheaves and line bundles. Gluing data. References: [5] Sheaf O(D). Interpretation of Pic(X) as a group of invertible sheaves. Rational map associated with linear system. Example: |O(1)| on P^1 |O(2)| on P^1 and plane conics |O(3)| on P^1 and canonical normal curves |O(2)| on P^2 and Veronese surface in P^4. |O(h-p)| on P^2 and rational projection P^2 -> P^1. References: [7] Lecture 6: Linar systems of curves on surfaces, elementary properties. Complete linear systems. Linear systems of plane lines and conics. Linear system of conics passing through n points on P^2. Rational maps associated with linear systems - classical (italian) definition. Cremona map. Geometry of the Cremona transformation. Reference: [7] Lecture 7: Geometry of the Cremona map around the origin (map (a,b) |-> (a,b/a) Blow-up of an affine plane in a point. Exceptional line. Embedding of a blow-up plane into A^2 x P^1. Interpretion of the plane blown up at a point as a moduli space (point, line). Reference: [5] Blow-up of A^3 at a point. Behaviour of singular plane curves under blow-up. Resolution of singulariyies of y^2 = x^3 - x, y^2 = x^3, y^2 = x^5. Successive blow-ups. Reference: [6] Lecture 8: Isolated and non-isolated singularities of surfaces. Examples. Classification of quadratic forms in 3 variables. Every non-degen. quadratic form is isomorphic to x^2+y^2+z^2. Resolution of x^2+y^2+z^2 = 0. Lecture 9: Resolution of D_4 by four blow-ups. Dynkin diagram of D_4. Definition of simple surface singularities (given as list). Intersection index of properly intersecting curves on a surface. Lecture 10: Intersection pairing is invariant with respect to linear equivalence, with a proof. Self-intersection of "movable" divisors. "Moving lemma" (statement only) and definition of self-intersection. Hodge index theorem - statement only. Mumford's theorem on the negative-definideness of the intersection form on the resolution of singularity. References: [9],[10] Lecture 11: Let C be a plane conic, S be a cone over C and S' be the resolution of S. Then S' is isomorphic to the total space of O_{P_1}(-2). If C is a plane rational curve of dergree n, then S' is the totakl space of O(-n). The following theorems were stated today, but not proved: 1. (-1) curves and their contraction. 2. Grauert-Mumford theorem on the negative definiteness of (C_i,C_j). 3. Classification of the negative-definite quadrartic forms and the ADE - theorem. Refernces: [9] Lecture 12: Classical invariant theory and simple singularities. 1. Platonic solids and classification of finite subgroups of SO(3). 2. Covering SL_2 -> SO(3) and lifting of the finite subgroups of SO(3). Binary groups of automorphisms of polyhedra. 3. The surface X/Gamma for finite subgroups of SL_2 and the invariant polynomials on C^2 w.r.t. Gamma. 4. Classification theorem (ADE) (statement only). Icosahedron and E_8. References: [11], [12] Lecture 13. 1. Felix Klein's description of the invariants of reflection groups. Computation of the surface singularity C^2/G, where G is a binary reflection group, for the groups of tetrahedron, octahedron and icosahedron: Solid Surface C^2/G Type of singularity Tetrahedron x^2+y^3+z^3 = 0 ?_4 Octahedron x^2+y^3+z^4 = 0 E_6 Icosahedron x^2+y^3+z^5 = 0 E_8 where z is the invariant obtained from the vertices, x is the invariant obtained from centers of edges, y is the invariant obtained from centers of sides. 2. Milnor number of a singularity - algebraic definition. Examples of computing the Milnor's number. References: [11]; Arnold's books. Lecture 14: Singularities of the form (*): z^2 = f(x,y). Branch curves. Theorem: ADE singularities are of the form (*). (Statement only). Idea of "canonical respolution". Fibered product. Example: z^2 = x^2 - y^3; resolution of the curve singularity leads to not-normal singularity. Integrally closed rings. References: [9], [14]. Lecture 15: (April 5, 2006) Ramification curves Normal rings and normalization Canonical resolution of z^2 = x^2 + y^3: fibered product, normalization, saddle y = uv. z^2 = x^2 + y^3 is of type A_1. Lecture 16: (April 26, 2006) How to glue projective varieties from affine charts. Differential forms on projective varieties. P^1 has no regular differential forms. Differential forms on elliptic curve x^3 + y^3 + z^3 = 0. Genus of a curve. Geometric definition (number of holomorphic forms) and topological definition. Lecture 17: (May 10, 2006): Genus of a curve - 2: Euler characteristic of a topological space. Two definitions: triangulations and Betti numbers. Riemann-Hurwitz formula for Euler characteristics, with proof. Example: x^3 + y^3 = z^3 on a complex plane is a torus. Lecture 18: (May 17, 2006): Hodge squares of an algebraic curve and algebraic surface. Irregularity and geometric genus of algebraic surfaces. For curves geometric genus coinsides with topological genus (Hodhe theorem)(statement only). Problem of computing the geometric genus of a surface in P^3. Adjunction formula - I. Lecture 19: (May 24, 2006): Invertible sheaves on varieties - reminder. Divisors, linear equivalence and invertible sheaves O(D). Cocycle construction of invertible sheaves. Canonical class of divisors on a variety. Canonical; class of P^2. Adjunction formula - II: two proofs of adjunction formula, via exact sequence ofsheaves of differential forms and via residues. Lecture 20 (June 5, 2006) Adjunction formula-3: adjunction for curves on P^2. Examples: Plane curves of degree 2,3,4. deg K_C = 2g-2, 2 computations: via adjunction formula on P^2, and via projection to P^1. Lecture 21: (June 12, 2006) Adjunction formula-4: adjunction for surfaces in P^3. Remark: Bertini theorem (with no proof). Surfaces of degrees <=3, 4, and >=5. Analogs with plane curves of degree <=2, 3 and >= 4. Example: theorem: quadric in P^3 is isomorphic to P^1 x P^1. The notion of rational surfaces, K3 and surfaces of general type. References: [8] Examples: Plane curves of degree 2,3,4. deg K_C = 2g-2, 2 computations: via adjunction formula on P^2, and via projection to P^1. Lecture 22: (June 19, 2006) 1. Linear systems with base points on P^2: Linear system |H - p| and projection from a point; Linear system |2H - p| and Veronese surface in P^4; Linear system |2H - p - q| and quadrics in P^3; Linear system |3H - p1 - ... -p6| and cubic surfaces in P^3. Counting of constants: every cubic surface in P^3 can be obtained in this way. References: [7] 2. Classes of curves of genus 0, 1, and >=2 and classes of surfaces of degree <=3, 4 and >=5 in P^3. Some comments on the classification of surfaces. ----------------------------------------------- ---------- References: [1] D. Mumford, The red book of varieties and schemes; [2] D. Mumford, Complex projective geometry. [3] R. Godement, Algebraic topology and theory of sheaves. [4] R. Bott, L. Tu, Differential forms in algebraic topology. [5] Griffiths, Harris. Principles of algebraic geometry. [6] Hartshorn, Algebraic Geometry. [7] Danilov, Algbraic varieties and schemes. (EMS volume Algebraic Geometry-1) [8] Shafarevich, Algebraic surfaces (review). (EMS volume Algebraic Geometry-2) [9] Barth, Peters, Van de Ven. [10] Mumford, Topology of normal singularities. [11] Felix Klein, Lectures on the icosahedron and solution of the equations of the fifth degree. [12] Felix Klein, lectures on the development of mathematics in the 19th century. [13] Arnold, ... [14] Zarissky, Samuel Commutative algebra.