
MAST90097 Algebraic Geometry

Semester II 2018 
Lecturer: Arun Ram, 174 Richard Berry, email: aram@unimelb.edu.au
Lecturer: Arik Wilbert, 219 Richard Berry, email: arik.wilbert@unimelb.edu.au
Time and Location:
Lecture: Wednesday 12:00  14:00 Peter Hall building Room 213
Lecture/Practice class: Friday 14:15  15:15 Peter Hall building Room 213
Arun Ram's consultation hours are Thursdays 14:0017:00
in Room 174 of Richard Berry.
Arik Wilbert's consultation hours are by appointment
Room 117 Old Geology South
The student representatives are
Quentin Bell email: qbell@student.unimelb.edu.au and
Tom Dove email: tdove@student.unimelb.edu.au
Announcements
 08.09.2018 The third assignment (due 12 October) is available
(pdf file available HERE)
 12.08.2018 The second assignment (due 7 September) is available
(pdf file available HERE)
 26.07.2018 The first assignment (due 10 August) is available
(pdf file available HERE)
 25.07.2018 Class is starting!
 The lectures will not be recorded.
 Prof. Ram reads email but generally does not respond by email.

Plagiarism declaration:
It is required to complete the
plagiarism declaration
in LMS, by clicking on the
Plagiarism declaration link for this subject and completing the submission
of the Plagiarism declaration through LMS for this subject.
 Academic Misconduct
The start of semester pack includes a statement about
Academic Misconduct (pdf file),
Further information regarding academic misconduct and plagiarism is available at
http://academichonesty.unimelb.edu.au/
 Generic Skills statement:
In addition to learning specific technical skills that will assist you in your future careers in science, engineering, commerce, education or elsewhere, you will have the opportunity to develop, in this subject, generic skills that will assist you in whatever your future career path:
 You will develop problemsolving skills (especially through tutorial exercises) including engaging with unfamiliar problems, and identifying relevant strategies.
 You will develop analytical skills  the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of the analysis.
 Through tutorials and other interactions with fellow students, you will develop the ability to work in a team. The school distinguishes between ethical collaboration, which is strongly encouraged, and plagiarism, which is prohibited.
 Printing arrangements for Peter Hall building
Students must use UNICARD to print documents. The UNICARD printer is located near the
G70 computer lab. Please note: there is not a money uploader in the Peter Hall Building. For more information about printing at the University and for locations of UNICARD
uploaders direct students to Student IT Support:
http://studentit.unimelb.edu.au/
 Housekeeping:
The start of semester pack includes:
Housekeeping (pdf file),
.
 Stop1 Quick Reference Guide pdf file
contains information on Requisite waivers, Enrolment after week 2, Electives, subject and major selection, study abroad, credit exemptions, academic adjustments, special consideration, exam reviews, Stop1 services, Healthand well being, further information.
 It is University Policy that:
“a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results. Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).
Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘makeup’ examination).
Subject Outline
The handbook entry for this course is at
https://handbook.unimelb.edu.au/view/2018/MAST90097.
This course is an introduction to algebraic geometry. Algebraic geometry is the study of zero sets of polynomials. It exploits the interplay between rings of functions and the underlying geometric objects on which they are defined. It is a fundamental tool in may areas of mathematics, including number theory, physics and differential geometry. The syllabus includes affine and projective varieties, coordinate ring of functions, the Nullstellensatz, Zariski topology, regular morphisms, dimension, smoothness and singularities, sheaves, schemes.
After completing this subject, students should gain:
 an understanding of the concepts of affine and projective varieties;

an understanding of schemes;

the ability to calculate invariants of varieties;

the ability to solve problems involving varieties by converting them into problems in algebra;

the ability to pursue further studies in this and related areas.
Assessment
Up to 60 pages of assignments (60%: three assignments worth 20% each, due early, mid and late in semester), a 2hour written examination (40%, in the examination period).
Note that each piece
of assessment is compulsory.
The goal of the assessment for this course
is to explain to the marker how well you
can navigate the material, not necessarily whether you succeed in solving
a question with a correct or incorrect answer.
The assessment will be marked accordingly.
"Proof machine" will be an important component of the presentation skills
to be improved in this course.
Assignments
Assignments will be due by 2pm on the following dates:
If you cannot get this assignment, or any other class materials from this site,
please ask the lecturer for a printed copy.
These assignments must be your own work. While students are encouraged to discuss
their coursework and problems with one another, assignments must be written up
independently. It is University policy that students submit a signed plagiarism
sheet at the start of each semester. If you do not submit this sheet your assignments
will be given a mark of zero.
 it is required to complete the plagiarism declaration
in LMS, by clicking on the
Plagiarism declaration link for this subject and completing the submission
of the Plagiarism declaration through LMS for this subject.
Students who are unable to submit an assignment on time and qualify for special
consideration should contact the lecturer as soon as possible after the due date.
Prerequisites
 MAST 30005 Algebra
 MAST90025 Commutative and Multilinear algebra
Additional References
The following additional references are recommended.
 G. Harder,
Lectures on Algebraic Geometry I and
Volume II,
(This is available online through Melbourne University library
first volume
and
second volume)
(2 volumes).
 I.G. Macdonald, Algebraic Geometry: Introduction to schemes, originally published by W.A. Benjamin, Inc 1968, avaliable from the page
http://math.soimeme.org/~arunram/resources.html
 A. Neeman, Algebraic and analytic geometry, Cambridge University Press 2007. This is available electronically from the University library
catalogue entry
 (This is in the High Use section of the ERC library)
Algebraic Geometry: Salt Lake City 2015 (Parts 1 and 2)
Proceedings of Symposia in Pure Mathematics
Volume 97, 2018 ISBN: 9781470446673
American Mathematical Society
https://bookstore.ams.org/pspum97/
 Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics Volume 52 (1977).

Igor R. Shafarevich, Basic Algebraic Geometry 1: Varieties in Projective Space, SpringerVerlag (1994).

David Eisenbud and Joe Harris, The geometry of Schemes, Springer (2000).
Recommended links from Arun Ram: Notes :
Lectures this semester
 Week 1 (Ram): What is a space?
handwritten lecture notes (pdf file)

points and relative position (topological spaces, metric spaces, polynomial,
continuous and differentiable functions)

functions and rings (affine varieties and affine schemes)

gluing (varieties, schemes, manifolds, CW complexes, simplicial sets)
 25 July 2018 Lecture 1:
sample exam questions (pdf file)
 27 July 2018 Lecture 2:
 Week 2 (Wilbert): Projective space
handwritten lecture notes (pdf file)
additional handwritten lecture notes (pdf file)
Friday handwritten lecture notes (pdf file)

quotient space, cell decomposition

structure sheaf,

line bundles, tangent space
 1 August 2018 Lecture 3:
sample exam questions (pdf file)
 3 August 2018 Lecture 4
 Week 3 (Ram): Flag varieties
handwritten lecture notes (pdf file)

classical flags, homogeneous space, cell decomposition

Structure sheaf, line bundles, tangent space, projective imbedding

Subvarieties, BottSamelson resolutions
 8 August 2018 Lecture 5:
sample exam questions (pdf file)
 10 August 2018 Lecture 6:
 Week 4 (Ram): More flag varieties and projective space
handwritten lecture notes (pdf file)

ℙ^{1} as a scheme and as CW complex

cell decomposition of flag varieties and flag variety as a scheme

Pic and global sections of line bundles
 15 August 2018 Lecture 7:
 17 August 2018 Lecture 8:
 Additional references
 J. Weinstein, Reciprocity
Laws and Galois representations: recent breakthroughs,
Bulletin of the American Mathematical Society, ????
 M. Mulase and M. Penkava, Combinatorial structure of the Moduli space of Riemann surfaces and the KP equations, 1997 manuscript.
 M. Hazewinkel and C.F. Martin,
A short elementary proof of Grothendieck's theorem on algebraic vector bundles over the projective line,
Journal of Pure and Applied Algebra 23 (1982) 207211.
 Week 5 (Wilbert): Elliptic curves
handwritten lecture notes (pdf file)

complex tori, elliptic functions, Pfunction

AppelHumbert theorem, projective embedding, homogeneous coordinate ring

modular forms
 22 August 2018 Lecture 9:
 24 August 2018 Lecture 10:
 Week 6 (Ram):
Vector bundles and Line bundles
handwritten lecture notes (pdf file)
 29 August 2018 Lecture 11:
 31 August 2018 Lecture 12:
 Week 7 (Ram):
Tangent space, connections and Dmodules;
handwritten lecture notes (pdf file)
 5 September 2018 Lecture 13:
 7 September 2018 Lecture 14:
 Week 8 (Wilbert):
Sheaf cohomology;
handwritten lecture notes (pdf file)
 12 September 2018 Lecture 15:
 14 September 2018 Lecture 16:
handwritten lecture notes (pdf file)
 Week 9 (Ram):
Abelian varieties
handwritten lecture notes (pdf file)

Siegel upper half space and polarizations

Theta functions

Siegel modular forms
 19 September 2018 Lecture 13:
 21 September 2018 Lecture 14:
handwritten lecture notes (pdf file)
 Week 10: Cohomology, Ktheory and GRR
handwritten lecture notes (pdf file)
 3 October 2018 Lecture 19:
 5 October 2018 Lecture 20: Examples of cohomology and Ktheory
handwritten lecture notes (pdf file)
 Week 11:
The BorelWeilBott theorem
handwritten lecture notes (pdf file)
 10 October 2018 Lecture 17:
 12 October Lecture 18:
 Week 12:
GAGA and Local systems and flat connections
handwritten lecture notes (pdf file)
 17 October 2018 Lecture 23:
 19 October 2018 Lecture 24:
Conversions between Weierstrass P function and Jacobi theta functions
 Week 13 (Ram):
Curves and M_{g,n}

genus, and basic RiemannRoch for curves

fat graphs, pairs of pants, and coordinatization

survey of topological recursion, 2D quantum field theories and cohomological field theories
 10 October 2018 Lecture 17:
handwritten lecture notes (pdf file).
 12 October Lecture 18:
handwritten lecture notes (pdf file).
 Week 14 (Wilbert): Singularities
 17 October 2018 Lecture 23:
handwritten lecture notes (pdf file).
 19 October 2018 Lecture 24:
handwritten lecture notes (pdf file).
 Week 15 (Ram):
Blowups, flips, flops and the minimal model program
 ?? October 2018 Lecture 21:
handwritten lecture notes (pdf file).
 ?? October 2018 Lecture 22:
handwritten lecture notes (pdf file).