# Computational Algebraic Geometry - Fall 2015

The lectures of this course will be based on a diverse selection of books and papers. However, the following books should constitute a great portion of the material for us.- Computations in algebraic geometry with Macaulay 2
- Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
- Using Algebraic Geometry
- Computational Algebraic Geometry

Download the coures outline here.

The course will be selfcontained. However, a solid background in undergraduate/graduate algebra is going to facilitate the process. Use

*Introduction To Commutative Algebra*by Atiyah and McDonald or

*Commutative Algebra with a View Toward Algebraic Geometry*by Eisenbud for the background. The asignments will be adjusted according to the level of the participating students.

The following list is an ambitious list of topics to include in this course. Although we might not be able to get through every topic, it's good to have a big picture to save us from getting trapped in the details.

- Introduction to polynomial systems, parametrizations, elimination and extension of solutions
- Groebner bases, monomial orderings, monomial ideals, Hilbert basis theorem, Buchberger's algorithm
- Affine and projective varieties, Nullstellensatz, primary decompositions
- Blowup and Rees algebras
- Dimension theory through Hilbert polynomials and series, graded algebras and modules
- Free resolutions, Koszul complexes, Hilbert-Burch theorem, Eagon-Northcott resolutions, regularity
- Complete intersection, Gorenstein and Cohen-Macaulay rings
- Hilbert schemes
- Applications and computations

## Projects, Presentations during class time, Tues/Fri at 4 PM

The written documents are downloadable through the following links.JERSON LEONARDO CARO REYES | Tues, Oct 27 |
Rees and Blowup Algebras, Intgeral closure and Numerical Invariants |

JUANITA DUQUE ROSERO | Fri, Oct 30 |
Newton Polytopes, Bernstein's Theorem, Discriminants and the Seconadary polytope |

JOSE MIGUEL CRUZ RANGEL | Tues, Nov 3 |
Real Algebraic Geometry, Positivity and Optimization |

GUSTAVO CHAPARRO SUMALAVE | Fri, Nov 6 |
The Riemann-Roch Formula |

JUAN RAFAEL ALVAREZ VELASQUEZ | Tues, Nov 10 |
Computer Vision and Hilbert Schemes |

DIEGO ANTONIO ROBAYO BARGANS | Fri, Nov 13 |
D-modules and the Bernstein-Sato Polynomial |

## Homeworks and stuff

HW #1HW #2

Midterm Test

HW #3

HW #4

HW #5

Final Exam

## Practice Homework

These are some suggested homework questions to get you started. This list will be updated as we make progress.From CLO | |

Read Chapter 1 | |

2.2 | 4, 5, 6, 7 |

2.3 | 2, 4, 5, 9 |

2.4 | 5, 8, 10, 11 |

2.5 | 7, 9, 11, 15, 18 |

2.6 | 2, 3, 5, 9, 11 |

3.1 | 1, 4, 5, 6, 7 |

3.2 | 3, 4 |

3.3 | 2, 3, 6, 8, 12 |

3.4 | 2, 4, 11 |

3.5 | 5, 8, 9, 10, 11, 16 |

3.6 | 1, 3, 4, 6, 8 |

4.4 | 3, 8, 9, 10 |

4.7 | 4, 6, 8, 11, 12 |

5.3 | 5, 7 , 9, 13 |

5.4 | 4, 5, 6, 9, 13, 14 |

8.2 | 13, 14, 16, 17, 18 |

8.3 | 4, 6, 10 |

8.4 | 9, 11, 12, 13, 14 |

9.2 | 7, 8, 11, 14 |

9.3 | 8, 9, 10, 16, 19, 20 |

Chapter 19 and 20 | Free resolutions and related topics: Commutative algebra with a view toward algebraic geometry by Eisenbud |

Section 4.6 | Local cohomology: An Intro to Local Cohomology by Weibel |

Chapter 6 | Grassmannians: Algebraic Geometry by Joe Harris |

Chapter 18 | Hilbert Schmes of Points on the plane: Combinatorial Commutative Algebra by Miller and Sturmfels |