Lectures: Tuesday, Thursdays and Saturdays 12:00 -13:00 at MR13. The revision class is on 22 May from 13:30 to 15:30 in MR13.

Announcements: The lecture that should have been on the 13th of February is moved to the 20th March at 12:00 in MR13.

Plan of the course

References 

• 1. W. Fulton Intersection Theory. 2nd edition. Springer. 

  2. D. Eisenbud and J. Harris, 3264 & All That. Intersection Theory in Algebraic Geometry. Cambridge University Press.

Supplementary material

1. Notes of the course

2. Exercises for the course

Schedule of the course

• Lecture 1 (23/01) : Review of Algebraic Geometry and Overview of the course (pdf)

• Lecture 2 (26/01): Orders of zeros and poles, rational equivalence, Pushfoward of cycles, Bézout's theorem, cycles of subschemes (pdf)

• Lecture 3 (28/01): Flatness and flat pullback (pdf)

• Lecture 4 (30/01): Excision sequence, flat pullback via affine bundles, Chow groups of schemes with a cellular decomposition, Chow groups of Grassmanians (pdf)

• Lecture 5 (01/02): Pseudodivisors, intersecting with divisors, first chern class of a line bundle (pdf)

• Lecture 6 (04/02): Gysin map for divisors, Segre and Chern classes of vector bundles, Splitting principle (pdf)

• Lecture 7 (06/02): Proof and applications of the splitting principle (pdf)

• Lecture 8 (08/02): Regular embeddings, Chern class of projective spaces, Adjunction formula (pdf)

• Lecture 9 (11/02): Riemann-Hurwitz formula, Chow groups of projective bundles (pdf)

• Lecture 10 (15/02): Generalities on cones (pdf)

• Lecture 11 (18/02): Segre class of cones, Segre classes of closed subschemes (pdf)

• Lecture 12 (20/02): Deformation to the Normal Cone (pdf)

• Lecture 13 (22/02): Specialization to the Normal Cone, Hilbert Polynomial (pdf)

• Lecture 14 (25/02): Degree of a closed subscheme of Projective space, Conics in P3 (pdf)

• Lecture 15 (27/02): 92 conics in P3 through 8 lines (pdf)

• Lecture 16 (01/03): Transversality statement in the enumeration of conics in P3 through 8 lines (pdf)

• Lecture 17 (04/03): Intersection products: the basic construction (pdf)

• Lecture 18 (06/03): Refined Gysin pullback, Excess intersection formula (pdf)

• Lecture 19 (08/03): Applications of the Excess intersection formula, introduction to Koszul complexes (pdf)

• Lecture 20 (11/03): Koszul Complexes, Generalities on Regular embeddings (with solutions to Exercise 3 Sheet 2) (pdf)

• Lecture 21 (13/03): Various functorial equalities of (Gysin) pullbacks, Local Complete Intersection Morphisms and their pullbacks (pdf)

• Lecture 22 (16/03): Monoidal transformations, Key formula, split exact sequences for Chow groups of Blow-ups (pdf)

• Lecture 23 (18/03): Blow-up formula and its Corollaries (pdf)

• Lecture 24 (20/03): Chow rings: projective spaces, projective bundles, Blow-ups of non-singular surfaces, products. Basis Theorem for Grassmanians (pdf)

• Revision class (22/05): Tour of the course and some exercises/examples: 27 lines in a cubic surface (pdf)

Exercises (curated by Terry) : Exercise sheet 1, Exercise sheet 2, Exercise sheet 3, Exercise sheet  4, Solutions sheet 1, Solutions sheet 2, Solutions sheet 3, Exercise sheet 4, Solutions sheet 4