## Introduction to Algebraic Geometry

We will develop the theory of algebraic varieties, which are zero sets of polynomial equations.
We will start with some basic commutative algebra - define Groebner basis, chain conditions and talk
about the ideal membership problem. Then we will discuss varieties - affine, projective, quasi-projective.
We will prove Hilbert's Nullstellensatz (one of the most important theorems of classical algebraic geometry).
We will talk about different notion of dimension and how those relate. We will define maps between
varieites - morphisms, rational and birational maps. The remaining topics are (but not limited to) singularity
theory, normalization and blow-ups, elimination theory, resultants, divisors on algebraic varieties. We will
try to also focus on some computational aspects of algebraic geometry.

### Course info

- Lectures: TTh 11:35-12:45
- Classroom: DL 431
- Office: DL 406

### Syllabus

You can find the syllabus

here.

### Office hours

If you need help with the course or you have questions regarding the material or the homeworks
or you would just want to pick up your homework, please don't hesitate to stop by my office or
write me an email. If the time is inconvenient you can arrange an appointment.

Office hours: Monday 5-6pm

### Homework Sets

### Some Macaulay2 code

- Examples: Groebner basis and division algorithm.

### Schedule

Week |
Dates |
Topics |

1 |
29 Aug - 1 Sept |
Introduction, monomial orders, division algorithm, introducing the ideal membership problem |

2 |
2 Sept - 8 Sept |
Dickson's lemma, Hilbert's basis theorem, existence of Groebner basis, Noether's proposition, Buchberger's criterion |

3 |
9 Sept - 15 Sept |
Buchberger's algorithm, examples, Macaulay2, varieties |

4 |
16 Sept - 22 Sept |
Hilbert's Nullstellensatz |

5 |
23 Sept - 29 Sept |
Coordinate rings, dimension, morphisms |

6 |
30 Sept - 6 Oct |
Rational and birational maps, Projective varieties |

7 |
7 Oct - 13 Oct |
Quasi-projective varieties, products of quasi-projective varieties |

8 |
14 Oct - 20 Oct |
The image of projective morphism is closed |

9 |
21 Oct - 28 Oct |
Families of varieties, lines on surfaces |

10 |
29 Oct - 3 Nov |
Elimination theory, resultants |

11 |
4 Nov - 10 Nov |
Tangent space |

12 |
11 Nov - 17 Nov |
Resolution of singulatities overview, curves, normalization |

13 |
24 Nov - 1 Dec |
Blow-ups |