# Basic Algebraic Geometry 2: Schemes And Complex...

I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with algebraic or projective geometry.

## Basic Algebraic Geometry 2: Schemes and Complex...

For a mixture of both, with a first half introduction to projective algebraic geometry and a second half heavily focused categorical introduction to schemes, this new book is a gem, and may be exactly what you are looking for, serving as a perfect introduction before/along with Hartshorne first chapters:

To clarify concepts on projective geometry, projective varieties and to supplement Hartshorne's reading, either from a complex geometry or purely algebraic point of view, the following long list of freely available online courses may provide you with the extra bits you need on specific topics (warning! most of them are more elementary than Hartshorne but some of them go beyond it or supplement it on other topics, they are included for completeness of good references to have if you decide to go beyond Hartshorne):

I found projective geometry confusing when I began learning algebraic geometry. Hartshorne has notes on projective geometry which are available online and which I found quite useful. Search Foundations of Projective Geometry by Robin Hartshorne online, or contact me via email (you will find my email address on my profile) and I will send you the notes.

Math 552 Algebraic Geometry I, Fall 2009 Welcome to Math 552! This course serves as an introduction to Algebraic Geometry. Algebraic Geometry is a central subject in modern mathematics, with close connections with number theory, combinatorics, representation theory, differential and symplectic geometry. We will study basic properties of projective algebraic varieties such as dimension, degree and singularities. At the same time, we will develop a large body of examples that motivate the study of the subject. Depending on time, we will develop the classical theory of curves and surfaces. This course should be enough preparation for a course on the theory of schemes. Lecturer: Izzet Coskun, coskun@math.uic.edu Office hours: M 11:30-12:30, W 9:30-11:30 and by appointment in SEO 423 Venue: Addams Hall 302 Time: 2:00-2:50 pm. Text book: In addition to course notes, there will be three recommended texts for this course. Joe Harris, Algebraic Geometry: a first course, Springer 1992.

Igor Shafarevich, Basic Algebraic Geometry I, Varieties in Projective Space, Springer-Verlag 1994.

David Mumford, Algebraic Geometry I, Complex Projective Varieties, Springer 1995.

Prerequisites: A first year graduate course in algebra: familiarity with commutative rings and modules. We will develop the necessary commutative and homological algebra in the course. Familiarity with differential geometry or topology helpful, but not required. The material covered in MATH 549 Differentiable Manifolds should nicely complement this course. Homework: There will be weekly homework. The homeworkis due on Wednesdays at the beginning of class. Late homework will notbe accepted. You may (in fact, you are encouaged to) work on problems togerther; however, thewrite-up must be your own and should reflect your own understanding ofthe problem. Grading: The grade will be entirely based on the homework. Additional references: The following is a list of references that are more advanced, but you might wish to consult them for more in depth treatments of the subject.

R. Hartshorne, Algebraic Geometry, Springer 1977.

D. Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics 1358, Springer-Verlag.

P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, 1978.

Course Notes Available on the Web: The following course notes are really nice. One is more basic and the other more advanced, but you might wish to refer to them.

This thesis consists of two parts, a first part on computations in algebraic geometry, and a second part on deformation quantization. More specifically, it is a collection of four papers. In the papers I, II and III, we present algorithms and an implementation for the computation of degrees of characteristic classes in algebraic geometry. Paper IV is a contribution to the field of deformation quantization and actions of the Grothendieck-TeichmÃ¼ller group.

1859: How many plane conics are tangent to five given conics? The solution to this problem will introduce many of the key tools in enumerative geometry, and in algebraic geometry in general. We will then do a brief tour of the related areas of physics and explore the connections between string theory and enumerative geometry.

These lectures will study the construction of quotients of algebraic varieties by algebraic group actions. As motivation, we will begin by discussing how the concept of a moduli space arises in classification problems in algebraic geometry. We will see that (when they exist) moduli spaces can often be constructed as quotients of algebraic varieties by group actions, but that only very special algebraic group actions on algebraic varieties have quotients which are themselves algebraic varieties in a natural way.

Next we will study the link between GIT and the concept of reduction in symplectic geometry using moment(um) maps, which leads to methods for studying the topology of GIT quotients and moduli spaces in complex algebraic geometry.

This semester I am teaching the course on schemes.If you are interested, please email me, tell me a littleabout yourself (academically), and I will add you to myemail list. The first lecture will be Jan 18 unfortunately on zoom.If I haven't email you about this yet, then email me, see above. The lectures will be Tuesday and Thursday 10:10am-11:25amLocation: 507 Mathematics Building TA: Noah Olander Office hours: 1 - 2:30 PM on Wednesdays. Exam. Thu, May 12 2022 9:00am-12:00pm in MATH 507 Very rough outline of the course. In the first few lecturesI will talk about the language of schemes. The idea is todiscuss this minimally, enough to introduce some of the conceptsneeded to discuss the topics below. Then I will turn the courseinto a sequence of mini-topics. I hope to say something about Moduli spaces: we can talk about moduli of varietiesand about moduli of vector bundles. Just enough materialso you have an idea what this even means. Coherent modules and their cohomology. This is a veryimportant technical tool, but also really fun in and of itself.I hope to be able to discuss some of the more fun aspects of the story. Rational points and heights. Rudimentary introduction.I personally wonder about the question: what is really requiredto have an adequate theory of heights? Tiny bit of intersection theory. We can discuss B\'ezout.We can talk about interscection theory on surfaces. We can discusswhy it is hard to get an intersection theory on chow groups of asmooth projective variety. Group schemes. What is a group scheme? Why do we not understandgroup schemes over the dual numbers? What is a representation of agroup scheme. Why is an elliptic curve a group scheme? What is anabelian variety? How do we prove Mordell-Weil using heights?Let me know if there are other basic topics you would be interested inhearing about.

TY - JOURAU - Kosarew, SiegmundTI - Geometric and categorical nonabelian duality in complex geometryJO - Annali della Scuola Normale Superiore di Pisa - Classe di ScienzePY - 2002PB - Scuola normale superioreVL - 1IS - 4SP - 769EP - 797AB - The Leitmotiv of this work is to find suitable notions of dual varieties in a general sense. We develop the basic elements of a duality theory for varieties and complex spaces, by adopting a geometric and a categorical point of view. One main feature is to prove a biduality property for each notion which is achieved in most cases.LA - engUR - ER -

116. Combinatorics. * Based on induction and elementary counting techniques: counting subsets, partitions, and permutations; recurrence relations and generating functions; the principle of inclusion and exclusion; Polya enumeration; Ramsey theory or enumerative geometry. Prerequisite(s): course 100. Enrollment restricted to sophomores juniors, and seniors. Familiarity with basic group theory recommended. The Staff

129. Algebraic Geometry. * Algebraic geometry of affine and projective curves, including conics and elliptic curves; Bezout's theorem; coordinate rings and Hillbert's Nullstellensatz; affine and projective varieties; and regular and singular varieties. Other topics, such as blow-ups and algebraic surfaces as time permits. Prerequisite(s): courses 21 and 100. Enrollment limited to 40. The Staff

239. Homological Algebra. * Homology and cohomology theories have proven to be powerful tools in many fields (topology, geometry, number theory, algebra). Independent of the field, these theories use the common language of homological algebra. The aim of this course is to acquaint the participants with basic concepts of category theory and homological algebra, as follows: chain complexes, homology, homotopy, several (co)homology theories (topological spaces, manifolds, groups, algebras, Lie groups), projective and injective resolutions, derived functors (Ext and Tor). Depending on time, spectral sequences or derived categories may also be treated. Courses 200 and 202 strongly recommended. Enrollment restricted to graduate students. The Staff

254. Geometric Analysis. S Introduction to some basics in geometric analysis through the discussions of two fundamental problems in geometry: the resolution of the Yamabe problem and the study of harmonic maps. The analytic aspects of these problems include Sobolev spaces, best constants in Sobolev inequalities, and regularity and a priori estimates of systems of elliptic PDE. Courses 204, 205, 209, 212, and 213 recommended as preparation. Enrollment restricted to graduate students. The Staff 041b061a72