MA986 Topics in Algebraic Geometry Timetable Mon 13:00 B3.03 from Mon 9th Jan 2023 Thu 13:00 in D1.07 except Week 5 Thu 9th Feb in B1.01 Fri 11:00 B3.02 The lectures are broadcast on my Zoom account Meeting ID: 857 140 6186 Password: 3MfD6v Module description ==== 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. ==== 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 ==== 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 * Johan de Jong and others, The Stacks Project https://stacks.math.columbia.edu/ ==== 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. ==== 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. ==== 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)