Course Identification

Introduction to Real Algebraic Geometry and Toric varieties

Lecturers and Teaching Assistants

Prof. Alexander Esterov

Course Schedule and Location

First Semester
Monday, 13:05 - 15:00, Ziskind, Rm 155
Wednesday, 13:05 - 14:15, Ziskind, Rm 1

Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; Elective; Regular; 2.00 points




You are expected to be familiar with linear algebra and geometry (including projective spaces and quadratic curves), point-set topology and analysis in Rn (including compactness, diffeomorphisms and submanifolds). Familiarity with convex polytopes and algebraic sets is a plus, however I can give a brief self-contained introduction sufficient for our course to the interested participants.



Language of Instruction


Attendance and participation

Expected and Recommended

Grade Type

Pass / Fail

Grade Breakdown (in %)


Evaluation Type

Final assignment

Scheduled date 1


Estimated Weekly Independent Workload (in hours)



The first part of the course will be devoted to Hilbert’s 16th problem for algebraic curves, which was one of the starting points for the real algebraic geometry. Given a smooth real algebraic plane curve of a given degree, how many components could it have? How many of them may be nested in the other ones?

Hilbert himself resolved the problem up to degree 6 modulo one elusive topological type, whose existence was proved only 70 years later. Our aim is Viro’s patchworking theorem, which allows to construct algebraic curves of a given degree with prescribed topology. For example, here is the patchworking construction for the aforementioned elusive curve:

The second part of the course will be devoted to toric varieties. We first enocunter them when proving the patchworking theorem, and they have many other applications accross various parts of geometry. Toric varieties are certain algebraic varieties that can be assigned to lattice polytopes. This correspondence between algebraic and polyhedral objects turns out to be extremely profitable for both fields of study. For instance, on the polyhedral side, it resolves the Upper bound conjecture on the number of faces of a simple polytope, and, on the side of algebric geometry, it leads to the theory of Newton polytopes.

Learning Outcomes

Understand the subject matter of real algebraic geometry.

Learn to apply patchworking techniques.

Understand the notion of a troic variety.

Be able to read research papers involving patchworking, projective toric varieties, and Newton polytopes.


Reading List

• V. Kharlamov and O. Viro, Easy reading on topology of real plane algebraic curves
• O. Viro, Patchworking Real Algebraic Varieties
• D. Cox, What is a Toric Variety?
• G. Ewald, Combinatorial Convexity and Algebraic Geometry

Most of these textbooks are freely available online.