Introduction to Enumerative Geometry

This is an intensive short course on enumerative geometry. The course covers an introduction to intersection theory, and applies the acquired techniques to some classical problems. We will introduce the basics of intersection theory: Chow ring, Chern classes, and basics of Schubert calculus. The theoretical tools which are developed will be applied to the enumerative geometry of some Grassmannian problem and to the Thom-Porteous formula for the calculation of the degree of determinantal varieties. If time permits, we will draw connections to the representation theory of the general linear group.

• Lecture 1, 2: Basics of intersection theory. Chow ring. Grassmannians.
• Lecture 3, 4: Chern classes. Schubert calculus. Enumerative problems.
• Lecture 5, 6: Thom-Porteous’s Formula. Representation Theory.

References

• D. Eisenbud, J. Harris 3264 and All That (Cambridge 2016) [main reference; lecture notes adapted from this reference will be provided (in non-final version)]
• E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris Geometry of Algebraic Curves, Vol. I (Springer 1985)
• L. Manivel Symmetric Functions, Schubert Polynomials, and Degeneracy Loci (SMF/AMS 1998)
• W. Fulton, J. Harris Representation Theory: A First Course (Springer 1991)

Date and time info
January 11 / 13 / 15 / 18 / 20 / 22, 2021: 16:30 - 18:30 CET

Prerequisites
A first course in algebraic geometry is recommended but not strictly required. Familiarity with the notion of algebraic variety and the Zariski topology in affine and projective space is assumed. Some familiarity with commutative algebra or algebraic topology will be helpful but not necessary.