# North German Algebraic Geometry Seminar WS 2019/20

## Schedule

Thursday 30.1.2020(lecture hall 1101-b305) | ||

14:15 - 15:15 | Kęstutis Česnavičius (Orsay) | Purity for flat cohomology |

15:15 - 15:45 | Coffee break | |

15:45 - 16:45 | Marc Paul Noordman (Groningen) | Autonomous first order differential equations |

17:00 - 18:00 | Khazhgali Kozhasov (TU Braunschweig) | On the number of critical points of a real form on the sphere |

19:00 h | Conference Dinner at restaurant Mikado, Schmiedestr. 3 |

Friday 31.01.2020(lecture hall 1101-b305) | ||

09:00 - 10:00 | Benjamin Schmidt (LU Hannover) | Derived Categories and the Genus of Curves |

10:00 - 10:30 | Coffee break | |

10:30 - 11:30 | Andrea Petracci (FU Berlin) | Smoothing toric Fano 3-folds |

11:45 - 12:45 | Hans-Christian Graf v. Bothmer (Hamburg) | Prelog Chow-Rings |

12:45 - 14:00 | Lunch break | |

14:00 - 15:00 h | Bruno Klingler (HU) | On the Zariski closure of the positive Hodge locus |

15:00 - 15:30 h | Coffee break | |

15:30 - 18:00 h | Retirement party | for Wolfgang Ebeling with talks by Michael Lönne (Bayreuth) and Sabir Gusein-Zade (Moscow) |

## Abstracts

**Prelog Chow-Rings**

Hans-Christian Graf v. Bothmer (Hamburg)

**Purity for the Brauer group of singular schemes**

Kęstutis Česnavičius (Orsay)

For regular Noetherian schemes, the cohomological Brauer group is insensitive to removing a closed subscheme of codimension ≥ 2. I will discuss the corresponding statement for schemes with local complete intersection singularities, for instance, for complete intersections in projective space. Such purity phenomena turn out to be low cohomological degree cases of purity for flat cohomology. I will discuss the latter from the point of view of the perfectoid approach to such questions. The talk is based on joint work with Peter Scholze.

**On the Zariski closure of the positive Hodge locus**

Bruno Klingler (HU Berlin)

Abstract: Given a variation of Hodge structures V on a smooth complex quasi-projective variety S, its Hodge locus is the set of points s in S where the Hodge structure V_{s} admits exceptional Hodge tensors. A famous result of Cattani, Deligne and Kaplan shows that this Hodge locus is a countable union of irreducible algebraic subvarieties of S, called the special subvarieties of (S,V). In this talk I will describe the geometry of the Zariski closure of the union of the positive dimensional special subvarieties. Joint work with Ania Otwinowska.

**On the number of critical points of a real form on the sphere**

Khazhgali Kozhasov (TU Braunschweig)

It is well-known that a generic real symmetric matrix of size *n* has exactly *n* real eigenvalues. Equivalently, a generic real quadratic form in *n* variables restricted to the unit sphere *S* has exactly *n* critical points. If *p* is a real form (homogeneous polynomial) of degree *d≥3*, the number *C*(*p*) of critical points of its restriction to the sphere *S* is not generically constant. In my talk I will describe typical values of the number *C*(*p*) that a generic *p* can attain.

**Autonomous first order differential equations**

Marc Paul Noordman (Groningen)

Autonomous algebraic first order differential equations (i.e. differential equations of the form P(u, u') = 0 for P a polynomial with constant coefficients) can be interpreted as rational differential forms on an algebraic curve. In this talk, based on joint work with Jaap Top and Marius van der Put, I will explain how this perspective clarifies the possible algebraic relations between solutions of such differential equations. In particular, we will see that there are very few algebraic relations between distinct non-constant solutions of the same differential equation, unless that differential equation comes from a one-dimensional group variety.

**Smoothing toric Fano 3-folds**

Andrea Petracci (FU Berlin)

In this talk I will show how to construct smoothings of toric Fano 3-folds with Gorenstein singularities starting from some combinatorial input on their associated polytopes. This result extends the toric dictionary between algebraic geometry and the combinatorics of polytopes. Moreover, this result fits into the context of Mirror Symmetry for Fano varieties. This is joint work with Alessio Corti and Paul Hacking.

**Derived Categories and the Genus of Curves**

Benjamin Schmidt (LU Hannover)

Abstract: A 19th century problem in algebraic geometry is to understand the relation between the genus and the degree of a curve in complex projective space. This is easy in the case of the projective plane, but becomes quite involved already in the case of three dimensional projective space. I will talk about generalizing classical results on this problem by Gruson and Peskine to other threefolds. This includes principally polarized abelian threefolds of Picard rank one and some Fano threefolds. The key technical ingredient is the study of stability of ideal sheaves of curves in the bounded derived category.