MA986 Topics in Algebraic Geometry
Timetable
Mon 13:00 B3.02 from Mon 9th Jan 2023
Thu 13:00 B3.02
Fri 11:00 B3.02
Module description
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Introductory description
Algebraic geometry has deep roots going back to Newton and Euler. The
modern subject has already been through many generations of increasing
sophistication and technical perfection.
The subject studies algebraic varieties, that are described as the
geometric locus defined by the vanishing of polynomial equations. That
slogan cuts both ways: in some cases a set of simultaneous equations is
the starting point, in search of a treatment using geometric ideas. Or
alternatively, a geometric construction raises questions in algebra as
to the best way of expressing it in terms of equations. For example, a
geometer may already have an algebraic curve in mind, and study how many
functions there are on it, and what algebraic equations they satisfy.
Algebraic geometry is one of the current growth areas of pure
mathematics, and it has deep and widespread influence in many other
areas of science: in pure mathematics (most obviously) geometry,
algebra, combinatorics, number theory; but also in interdisciplinary
contexts related to different flavours of theoretical physics.
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Outline syllabus for publication
The lecturer has an extremely large choice of topics, and we expect that
over the years, different lecturers will cover widely different topics.
We list some currently prominent directions that could be covered, but
this is not intended to be comprehensive.
* Ideas of toric geometry
* Quotient of a variety by a finite group
* Cyclic quotients singularities and their resolution
* The resolution of surface singularities
* Divisors and line bundles on curves and surfaces and higher
dimensional varieties; Cartier divisors versus Weil divisors
* The Picard group of line bundles up to linear equivalence
* Locally free sheaves and vector bundles
* Rank 2 vector bundles on a curve and elementary transformations of
ruled surfaces
* First notions of coherent sheaves and their cohomology
* First examples of algebraic surfaces
* The degree of a line bundle restricted to an algebraic curve in a
variety, and the pairing between Pic S and the homology group H_2(S) or
the N\'eron-Severi group N_1(S)
* Local intersection numbers of curves on a nonsingular surface
* B\'ezout's theorem for curves in PP^2
* Intersection numbers of curves on a projective surface
* The canonical class K_S and the adjunction formula
* First examples of moduli spaces, esp. Hilbert schemes and the Picard
variety as moduli space of line bundles up to isomorphism
* Geometric invariant theory and methods of constructing moduli spaces
* Algebraic stacks
Topics in algebraic surfaces:
* contraction of -1-curve
* rational and ruled surfaces. Tsen's theorem
* Extremal rays and minimal models via Mori theory,
* Birational transformations of surfaces and the Cremona group
* Canonical curves and K3 surfaces
* Elliptic surfaces
* The proof of the classification of surfaces via the classical method
of adjunction terminates
* The classification of surfaces via Mori theory
* Graded ring methods
* Examples of surfaces of general type with small invariants
* The Kodaira-Bombieri theorems on projective embeddings of surfaces of
general type
* Basic ideas of Hodge theory
* Moduli and periods of K3 surfaces
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Indicative reading list
* M. Reid, Undergraduate Algebraic Geometry, London Math. Soc. Student
Texts 12, Cambridge University Press (2010)
* J. Harris, Algebraic Geometry, A First Course, Graduate Texts in
Mathematics 133, Springer-Verlag (1992)
* I.R. Shafarevich, Basic Algebraic Geometry 1, third edition, Springer
(2013)
* R. Hartshorne, Algebraic geometry, Springer Graduate Texts, No. 52
(1977)
* Ravi Vakil, The Rising Sea, Foundations of Algebraic geometry (find
online)
* M. Reid, Chapters on algebraic surfaces, Complex algebraic geometry
(Park City, UT, 1993), 3-159 (free copies available). More material
online http://www.warwick.ac.uk/~masda/ + Algebraic geometry links
* J. Koll{\'a}r and S. Mori, Birational geometry of algebraic varieties,
Cambridge Tracts No 134, CUP 1998
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Principal module aims
The course will take PhD students from a modest background in algebra,
geometry and topology, and introduce them to the ideas of algebraic
geometry, including the technical background in commutative algebra,
scheme theory and cohomology, but also the rich theory of algebraic
curves and algebraic surfaces and beyond. The ideas here feature a rich
corpus of examples and methods for working with them.
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Learning outcomes
The successful student will take on board the technical ideas of modern
algebraic geometry, including schemes and cohomology. They will acquire
facility in handling the technical tools and many of the prominent
objects of study in geometry. They will also become aware of several of
the recent developments in higher dimensional geometry, including the
work of several recent Fields Medallists.
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Subject specific skills
Appreciation of ideas from different categories of geometry and their
interaction. For example: (1) the topological and algebraic view of
vector bundles. (2) Algebraic and complex analytic approaches to
cohomology and the Riemann-Roch. (3) Scheme theoretic synthesis of
areas of algebraic number theory and algebraic geometry.
Expertise in higher dimensional geometry, one of the current growth
areas in pure mathematics. The ideas of higher dimensional birational
has many applications to representation theory and theoretical physics,
notably aspects of string theory and quantum field theory.
Transferable skills
* sourcing research material
* prioritising and summarising relevant information
* absorbing and organizing information
* presentation skills (both oral and written)
Description
An oral exam involving a presentation by the student, followed by
questions from the panel (2 members of the department)
Methods for providing feedback on assessment
Students will receive feedback from the course instructor after the oral
exam, to cover also areas like presentation skills and use of
technologies (or blackboard)