# «Algebraic Geometry»

Instructor: Mathias Schulze
Assistant: Lucas Ruhstorfer
Lecture: Tue./Thu. 8:15-9:45am, room 48-438
Tutorial: Wed. 3:30-5:00pm, room 11-241

## Topics

Algebraic geometry is concerned with the study of solutions of polynomial equations. As opposed to linear algebra, such solutions can usually not be described explicitly. Instead, algebraic geometry provides an abstract language and tool box to understand polynomial equations from a qualitative point of view, e.g. to make sense of the dimension of the solution space. These methods are based on the interplay of (commutative) algebra and geometry/topology.

The main objects of interest are so-called varieties. These are zero sets of polynomials equiped with the Zariski topology. A basis of open sets is given by the non-vanishing of polynomials. There is a structure sheaf containing all polynomial functions on various open sets of the variety.

The course is an introduction to the basic concepts of algebraic geometry covering affine/projective varieties and morphisms, dimension, singularities and blowup, divisors on curves, Bézout's theorem with applications, and schemes.

## Tutorial

The tutorial will be held by Lucas Ruhstorfer. If you wish to attend, please enroll via URM by April 24, 2015.

## Homework

There will be weekly homework assignments posted on this page. Typically assignment will be posted on Friday, and are due on the following Friday by 11:00am. Please drop your solutions by the respective due date in the mail box of Lucas Ruhstorfer next to room 48-210. You may (and you are encouraged to) turn in your homework in teams of two. However, each of you needs to be able to present your team's solutions during the tutorial.

## Credit

In order to obtain an Übungsschein you need to actively participate in the tutorial and achieve a total homework score of at least 40%. Active participation means regular attendance and presenting a solution of a homework problem (at least once). For credit points you need to pass an oral exam after the end of the semester.

## Literature

• I. Dolgachev, Introduction to Algebraic Geometry, Lecture Notes (2013)
• A. Gathmann, Algebraic Geometry, Class Notes (2014)
• R. Hartshorne, Algebraic Geometry, Springer Graduate Texts in Mathematics 52 (1977)
• J. Harris, Algebraic Geometry, Springer Graduate Texts in Mathematics 133 (1992)
• D. Mumford, The Red Book of Varieties and Schemes, Springer Lecture Notes in Mathematics 1358 (1988)
• H.A. Nielsen, Algebraic Varieties, Lecture Notes (2003)
• M. Reid, Undergraduate Algebraic Geometry, London Mathematical Society Student Texts 12, Cambridge University Press (1988)
• I. Shafarevich, Basic Algebraic Geometry I - Varieties in Projective Space, Springer (1994)
• I. Shafarevich, Basic Algebraic Geometry II - Schemes and Complex Manifolds, Springer (1994)

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