Lectures: Mo 16:00-17:30 Geom H3 and Fr 10:15-11:45 Sed 19, 203

Excercises: Mo 17:45-19:15 Geom H3

We want to introduce to algebraic geometry so that we understand at least some basic facts necessary for the construction of the Hilbert scheme of points on an algebraic surface. A first course in algebra is a mandatory prerequisite for this course. Further knowledge of advanced algebra, function theory, differential geometry may occasionally be helpful. We start with a quick presentation of classical algebraic geometry and then focus to the modern approach using the language of schemes. There are many good books on algebraic geometry and even more different flavours

Math overflow question: best algebraic geometry textbook?

In this course we follow mostly these lecture notes:

Introduction to Schemes. G. Ellingsrud, J.C. Ottem

Other accessible books (tbc):

A Royal Road to Algebraic Geometry, A. Holme

Week 1: Informal introduction (cf. above links)

Week 2: EO pp. 1-15 classical algebraic geometry

Week 3: EO pp. 15-26 prime spectrum, Zariski topology, residue fields, generic point

Week 4: EO pp. 26-41 affine spaces, irreducibility, connectedness, distinguished open sets, maps of prime spectra, fibres

Week 5: EO pp. 41-61 (pre-)sheaves, morphisms of (pre-)sheaves, stalks, locally ringed space

Week 6: EO pp. 53- affine schemes,...

Week 7/8: EO pp. -- 100 glueing schemes, examples

Week 9/10: EO pp. -- 128 finitness conditions, dimension

Week 11/12: Hilbert schemes of points

Additional literature:

A. Ritter, Introduction to schemes

R. Borcherds, Lectures on algebraic geometry

R. Borcherds, Lectures on schemes

L. Tu, An introduction to manifolds

03.05.2024: structure sheaf and locally ringed spaces

J. Neukirch: Algebraische Zahlentheorie

D. Perrin: Algebraic geometry

D.Eisenbud, J. Harris: The geometry of schemes

27.05.2024: Mathematics in Lean

R.F. Mir: Schemes in Lean

K.Buzzard: GROTHENDIECK'S USE OF EQUALITY , K. Buzzard: The future of mathematics

17.06.2024: Hilbert Schemes of points

D. Bejleri: HILBERT SCHEMES: GEOMETRY, COMBINATORICS, AND REPRESENTATION THEORY

E. Miller: Appendix: Hilbert schemes of points in the plane, E. Miller, B. Sturmfeld: Combinatorial Commutative Algebra, Chapter 18

M. Skjelnes: HILBERT SCHEMES OF POINTS ON SMOOTH SURFACES