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MA940 - Coherent Sheaves and Cohomology

Lecturer: Miles Reid

Term: Term 2, that is, 10 weeks 11th Jan-Fri 19th Mar 2021

Commitment: 30 lectures. Provisional timetable: Mon 16:00, Thu 12:00, Fri 10:00
on Teams (email me to sign up).

More: See my MA940 webpage for more material, including videos of past lectures

Assessment: Oral exam (only available for credit to UoW PhD students)

Prerequisites: See below

Content: The main trunk of the lecture course is the classic:

[FAC] Jean-Pierre Serre, Faisceaux algébriques cohérents
Annals of Mathematics (2) 61:2 (1955) 197-278.

Download it from JSTOR https://www.jstor.org/stable/1969915

There is even an English translation for the less pretentious.

Serre's classic paper is to some extent dated, although the historical context
is valuable, as I will discuss. Hartshorne's book (appearing 20 years later) has
some technical improvements, but has its own difficulties for the beginning
student. I adopt Serre's paper as the main trunk, to build the subject on the
simplest possible foundations, with many side branches treating different points
of view: some historically important, providing intuitive alternatives to
illustrate the bigger picture, relating the material to other branches of
geometry, or (as time allows) going further into foundational points or
technical developments.

A light-weight treatment of many of the ideas of the course is contained in
Chapter B of my Park City Chapters on Algebraic Surfaces. My intention there was
a colloquial summary of the theory and how to use it, rather than any systematic
development of the material. I consider it reasonable to assume that any student
interested in this module will be willing to skim through my Chapter B to get
the flavour of the material.

References: Robin Hartshorne, Algebraic geometry, Graduate Texts, Springer, 1977.
(Warwick library e-book: http://webcat.warwick.ac.uk/record=b3214992~S1)

Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry,
http://math.stanford.edu/~vakil/216blog/

Miles Reid, Chapters on algebraic surfaces, in Complex algebraic geometry
(Park City, UT, 1993), 3-159 (1997). Printout will be available from shelves
outside B2.30. The preprint arXiv: alg-geom/9602006 is on my website (but
the published version is nicer to read).

If time allows, I may also cover the key papers
Armand Borel and Jean-Pierre Serre, Le théorème de Riemann-Roch, Bull.
Soc. Math. France 86 (1958), 97-136
Alexander Grothendieck, La théorie des classes de Chern, Bull. Soc. Math.
France 86 (1958), 137–154
See also Raoul Bott's detailed review MR0116022 (22 #6817 and #6818)