# Diophantine Geometry Notes And Homework

Some of the theorems presented in lecture will be demonstrated using the Sage computer algebra system, which is based on Python™. You can download a copy of Sage to run on your own machine if you wish, or create an account for free on the SageMathCloud™.

LEC # | TOPICS |
---|---|

1 | Introduction to Arithmetic Geometry (PDF), 18.782 Lecture 1 (SWS) |

2 | Rational Points on Conics (PDF) |

3 | Finite Fields (PDF), 18.782 Lecture 3 (SWS) |

4 | The Ring of p-adic Integers (PDF) |

5 | The Field of p-adic Numbers, Absolute Values, Ostrowski's Theorem for Q (PDF) |

6 | Ostrowski's Theorem for Number Fields (No lecture notes but see Ostrowski's Theorem for Number Fields (PDF) by Keith Conrad) |

7 | Product Formula for Number Fields, Completions (PDF) |

8 | Hensel's Lemma (PDF) |

9 | Quadratic Forms (PDF) |

10 | Hilbert Symbols (PDF), 18.782 Lecture 10 (SWS) |

11 | Weak and Strong Approximation, Hasse-Minkowski Theorem for Q (PDF) |

12 | Field Extensions, Algebraic Sets (PDF) |

13 | Affine and Projective Varieties (PDF) |

14 | Zariski Topology, Morphisms of Affine Varieties and Affine Algebras (PDF) |

15 | Rational Maps and Function Fields (PDF) |

16 | Products of Varieties and Chevalley's criterion for Completeness (PDF) |

17 | Tangent Spaces, Singular Points, Hypersurfaces (PDF) |

18 | Smooth Projective Curves (PDF) |

19 | Divisors, The Picard Group (PDF) |

20 | Degree Theorem for Morphisms of Curves (PDF) |

21 | Riemann-Roch Spaces (PDF) |

22 | Proof of the Riemann-Roch Theorem for Curves (PDF) |

23 | Elliptic Curves and Abelian Varieties (PDF) |

24 | Isogenies and Torsion Points, The Nagell-Lutz Theorem (PDF) |

25 | The Mordell-Weil Theorem (PDF) |

26 | Jacobians of Genus One Curves, The Weil-Chatelet and Tate-Shafarevich Groups (PDF) |

Mircea Mustaţă

I am a Professor in the Department of Mathematics at University of Michigan. I am supported by the NSF and by a Packard Fellowship. Here is my contact information. My work is in algebraic geometry. In the past few years my main interest was in various invariants of singularities of algebraic varieties, such as minimal log discrepancies, log canonical thresholds, multiplier ideals, Bernstein-Sato polynomials and F-thresholds. Various points of view and techniques come in the picture when studying these invariants: resolutions of singularities, jet schemes, D-modules or positive characteristic methods. Other interests include birational geometry, asymptotic base loci and invariants of divisors, and toric varieties. Here are my CV and a list of publications. ## PapersYou can find here my papers on the archive. Here are a few recent ones: |

1. The combinatorics and topology of proper toric maps (with Mark de Cataldo and Luca Migliorini), available at arXiv:1407.3497.

2. The dimension of jet schemes of singular varieties, available at arXiv:1404.7731.

3. Weight functions on non-archimedean analytic spaces and the Kontsevich-Soibelman skeleton (with Johannes Nicaise), available at arXiv:1212.6328.

4. On the numerical dimension of pseudo-effective divisors in positive characteristic (with Paolo Cascini, Christopher Hacon, and Karl Schwede), available at arXiv:1206.6521.

5. An algebraic approach to the openness conjecture of Demailly and Kollar (with Mattias Jonsson), available at arXiv:1205.4273.

6. A Frobenius variant of Seshadri constants (with Karl Schwede), available at arXiv:1203.1081.

7. The augmented base locus in positive characteristic (with Paolo Cascini and James McKernan), available at arXiv:1111.3236.

8. The non-nef locus in positive characteristic, available at arXiv:1109.3825.

## Teaching

Math 711. Topics in birational geometry.

Math 537. Introduction to differential topology.

Math 731. Spaces of arcs and singularities in birational geometry.

Math 732. Introduction to birational geometry.

Math 732. Introduction to diophantine approximation on abelian varieties. Mitya Boyarchenko kindly posted scanned versions of his notes here.

Math 732. Zeta functions in algebraic geometry. Here are the lecture notes from the course.

Math 632. Algebraic geometry II.

Math 631. Algebraic geometry I.

Math 631. Algebraic geometry I.

Math 632. Algebraic geometry II (Schemes and cohomology). Here are the homework assignments and the problems covered in the discussion session.

Math 731 and 732. Topics in Algebraic Geometry I and II (Toric varieties). Here are the lecture notes, though some chapters are still missing. The first chapters are just expanded versions of the corresponding chapters in Bill Fulton's book "Introduction to toric varieties", using also Bill's lecture notes for a course he taught a few years ago.

## Other activities

1. Tommaso de Fernex, Brendan Hassett, Martin Olsson, Mihnea Popa, Richard Thomas, and I are planning a successor to Seattle 2005 (and Santa Cruz 1995, Bowdoin 1985, Arcata 1974, Woods Hole 1964). This will happen between July 13-31, at University of Utah.2. Claude Sabbah, Christian Schnell, and I organized a workshop on Saito's theory of mixed Hodge modules, at the Clay Mathematics Institute, in Oxford (UK), between August 19-23, 2013.

3. Marian Aprodu, Mihnea Popa, and I organized a special session on algebraic geometry at the Joint International Meeting of the AMS and the Romanian Mathematical Society, in Alba Iulia (Romania), between June 27-30, 2013.

4. Nero Budur, Francois Loeser, and I organized a thematic program Motivic invariants and singularities at Notre Dame, between May 20-June 7, 2013.

5. Christopher Hacon, Mihnea Popa, and I organized an algebraic geometry conference between May 16-19, 2013, in Ann Arbor, in honor of Rob Lazarsfeld's 60th birthday.

6. Craig Huneke, Yujiro Kawamata, Karen Smith, Kei-ichi Watanabe, and I organized the workshop The Commutative Algebra of Singularities in Birational Geometry: Multiplier Ideals, Jets, Valuations, and Positive Characteristic Methods at MSRI, between May 6-10, 2013, as part of the Commutative Algebra program in 2012-2013.

7. I gave a micicourse on "Invariants of singularities" at the IMPANGA Summer School in Algebraic Geometry, held between July 4-10, 2010 at Bedlowo, Poland. Here are the lecture notes.

8. David Eisenbud, Craig Huneke, Claudia Polini, and I were the mentors of the MRC program in Commutative Algebra, held at Snowbird, Utah between June 26-July 2, 2010.

9. Lucia Caporaso, Brendan Hassett, James McKernan, Mihnea Popa, and I organized the introductory MSRI workshop Classical algebraic geometry today, between January 26-30, 2009. This was part of the MSRI program in algebraic geometry in Spring 2009.

10. Jeff McNeal and I organized a program in Park City "Analytic and algebraic geometry: common problems-different methods", between July 7-25, 2008.

11. In August 2006 Stefan Kebekus and Thomas Eckl organized a Summerschool in Cologne on "Moduli spaces and arcs in birational geometry". I was one of the lecturers and working group mentors (the other two were Gabi Farkas and Mihnea Popa). Here are my lecture notes.

## Some events in Ann Arbor

1. In Winter 2015 we will have two series of Spring Lectures in Algebraic Geometry. The speakers will be Peter Scholze and Jacob Lurie.

2. Between May 26-June 22, 2014 we had a special month dedicated to birational geometry in zero and positive characteristics.

## Future and past conferences

1. The “Algebraic Geometry” workshop, Oberwolfach, March 15-21, 2015.2. Georgia Algebraic Geometry Symposium, Athens, October 17-19, 2014.

3. The International Congress of Mathematicians, Seoul, August 13-21, 2014.

4. The conference “Commutative Algebra and Singularity Theory 2014”, in honor of Kei-ichi Watanabe, Toyama, July 28-August 1, 2014.

5. The “Classical algebraic geometry” workshop, Oberwolfach, June 30-July 4, 2014.

6. The workshop “Birational geometry and foliations”, Bonn, February 24-28, 2014.

7. The conference Géométrie birationnelle des variétés algébriques complexes, in honor of Frederic Campana's 60th birthday, Luminy, October 7-11, 2013.

## Photos

Here are some pictures taken in 2008, while hiking in Utah. mmustata@umich.edu

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