Tuesdays 14:20-16 and Fridays 9:20-11 in Aula Seminario II, starting on Tue, Mar 4, until Fri, Mar 28.
On Friday, Mar 14 lecture cancelled due to severe weather alert.
On Friday, Mar 28 we meet at 14:15 in Aula Tonelli.
On Tuesday, Apr 1 we meet at 14:15 in Aula Seminario VIII Piano.
Books/notes:
W. Fulton, "Intersection theory"
D. Eisenbud and J. Harrris, "3264 & All That - Intersection Theory in Algebraic Geometry"
R. Vakil, "Introduction to intersection theory in algebraic geometry"
Recordings:
J. Harris, "Intersection theory" @Escuela CIMPA-ICTP-UNESCO-MICINN-Santaló de Geometría Algebraica y Aplicaciones. Buenos Aires, 2011
P. Aluffi, "Segre classes and other intersection-theoretic invariants of singular schemes" @the Worldwide Center of Mathematics, 2018
Further reading:
J. Kock, I. Vainsencher, "An invitation to quantum cohomology"
J. Milnor and J. Stasheff, "Characteristic classes"
A. Ricolfi, "An Invitation to Modern Enumerative Geometry"
Syllabus:
04/03: Motivation: the number of rational plane curves of degree d through 3d-1 general points. Enumerative geometry=(compact) moduli spaces+intersection theory.
Remarks on Bèzout's theorem. Rational equivalence for divisors (motivated in the case of Cartier divisors) and higher codimension cycles. Moving lemmas: Kleiman's theorem for homogeneous spaces, and the (non-)example of the quadric cone. Comparison with (co)homology: Chow groups are large, e.g. A_0(curve of genus g) contains an Abelian variety of dimension g, and A_0(surface with p_g>0) is not even of finite type (after Mumford).
07/03: Example: the Chow ring of affine space. Functoriality toolkit: excision, Mayer-Vietoris, proper pushforward, flat pullback, pullback between smooth varieties. The pullback is a ring homomorphism, for f:X-->Y the push-pull formula makes A*(X) into a graded A*(Y)-module. Quasi-affine stratifications. Example: the Chow ring of projective space. Some remarks on flatness, the Hilbert polynomial and the degree of a homogeneous ideal, the cycle of a subscheme, and why only the dimension and the degree of a subvariety are visible from its cycle class.
11/03: Example: the Chow ring of a product of projective spaces, and the class of the diagonal. The degree of the Veronese and the Segre embeddings.
Introduction to Grassmannians: the Plücker embedding (coordinates and relations), affine charts and identification of the tangent space at W<V with Hom(W,V/W). The Grassmannian G(1,3) of lines in P^3. Schubert varieties give an affine stratification of G(1,3), hence their classes generate the Chow group. Calculation of the multiplicative structure.
18/03: A presentation of A^*(Gr(2,4)). There are 2d^4 lines meeting four general curves of degree d in P^3. A survey of A=A^*(Gr(k,n)): Young diagrams (A has rank {n \choose k}), duality and Littlewood-Richardson coefficients, Pieri's and Giambelli's formulae (A is multiplicatively generated by special Schubert classes), Whitney's formula.
Introduction to Chern classes: as degeneracy loci of sections, when the bundle is globally generated. Axiomatic approach.
21/03: Chern classes of tautological bundles on Grassmannians as special Schubert classes, Whitney's formula again. The splitting principle for Chern classes.
The Fano scheme of k-planes on a projective hypersurface: expected dimension and Chow class (as vanishing locus of a tautological section). There are 27 lines on a general cubic surface. The tangent space to the Fano scheme at [L] is H^0(L,N_{L/X}). The normal bundle exact sequence: any line on a smooth cubic surface is a (-1)-curve, in particular it is an isolated and reduced point of the Fano scheme.
25/03: conics tangent to a given line (resp. plane curve of degree d) form a hypersurface of degree 2 (resp. d^2+d) in P^5, but the number of conics tangent to five given line in general position is 1, not 32, as one can see from projective duality. The problem with Bèzout is that all these hypersurfaces intersect along the Veronese surface S of double lines. We discussed projective duality for conics, and how the blow-up of P^5 in S can be given a modular interpretation in terms of completed conics (C,C*).
28/03: Let D_1,...,D_5 be conics in general position and let Z_i in P^5 be the space of conics tangent to D_i (a degree 6 hypersurface), tZ_i its strict transform in the space of completed conics. Then tZ_1,...,tZ_5 may only intersect in the locus of smooth conics. Moreover, they intersect transversely (a general C will be tangent to D_i at a single point p, and the tangent space of Z_i in [C] can be identified with the space of conics passing through p). We count the intersection points (3264) by studying the intersection ring of the space of completed conics.
Returning to the question of conics tangent to five general lines, we study the problem by comparing the original intersection product Z_1...Z_5 with the product of the strict transforms tZ_i=Z_i-E, where E denotes the exceptional divisor of the blow-up. The class E-E^2+... appears naturally: we call it the Segre class (of the normal bundle of the Veronese surface S in P^5). The fact that Segre=1/Chern allows us to compute the contribution of S to the intersection product, which is a class of degree 31.
01/04: Review of the excess intersection formula. As an application we show that the twisted cubic can be written as the intersection of three surfaces in P^3 if and only if they are quadrics. For globally generated vector bundles , we interpret Segre classes (up to sign) as degeneracy loci, where r+i-1 general sections fail to generate E. We come back to the splitting principle and the construction of Chern classes: a key observation is that the intersection product with a Cartier divisor D can be refined to give a map A_k(X)->A_{k-1}(D). We state a description of the Chow ring of a projective bundle (generated over the base by the class of O(1) with a single relation where the Chern classes of E appear as coefficients) and a vector bundle (homotopy invariance: flat pullback is an isomorphism with inverse given by the pullback along the zero section).
We review the definition of multiplicity of a rational function along a divisor as approximating it with its leading term. This brings to the notion of normal cone. We define the multiplicity of a variety Y at a subscheme X both algebraically (from the leading coefficient of the Hilbert polynomial h(d)=dim(O_{Y,X}/m^d)) and geometrically (p_*(c_1(O(1))^{c-1} \cap [PC], where c=codim(X,Y), or equivalently by self-intersecting the exceptional divisor of Bl_X(Y)). More generally, we define the Segre class of X in Y.
Finally, we define the intersection product with a regularly embedded subscheme X of codimension c in Y: if V is a cycle on Y (or more generally a variety with a morphism to Y) and W its intersection with X (fibre product), the intersection product of V with X is a cycle on W obtained by approximating W with its normal cone C_{W/V}, and then intersecting the latter with the zero section of N_{X/Y|V}. In formulae, [X][V]={c(N_{X/Y}\cap s(W,V)}_{dim(V)-c}. The intersection product on a smooth variety Y comes from viewing it diagonally embedded in YxY (the conormal sheaf of the diagonal is the sheaf of Kähler differentials of Y).