*Humboldt-Universität zu Berlin *

*Mathem.-Naturwissenschaftliche Fakultät*

*Institut für Mathematik*

Sommersemester 2018

Das Forschungsseminar findet mittwochs in der Zeit von 13:00 - 15:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 1.114, statt.

Seminar: Algebraic Geometry an der FU

25.04.2018 | Einstein Lectures in Algebraic Geometry: Rahul Pandharipande (ETH Zürich) | |

Title: On Lehn's conjecture for Segre classes on Hilbert schemes of points of surfaces and generalizations |
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Abstract: Let L->S be a line bundle on a nonsingular projective surface.
I will discuss recent progress concerning the formula conjectured by Lehn in 1999
for the top Segre class of the tautological bundle L^[n] on Hilb(S,n) and
the parallel question for vector bundles V->S. Results of Voisin play a crucial role.
The talk represents joint work with A. Marian and D. Oprea.
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02.05.2018 | Mara Ungureanu (HU Berlin) | |

Title: The geometry of De Jonquières divisors on algebraic curves |
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Abstract: De Jonquières divisors are divisors with prescribed multiplicity contained in a linear series on a smooth projective curve. Viewing the curve as embedded in projective space by the linear series, the points in the support of these divisors determine hyperplanes that intersect the curve with prescribed multiplicities at the points of intersection. In this talk we give a broad overview of many interesting results pertaining to this setup, starting from de Jonquières' classical enumerative formula from 1866, and up to contemporary results involving the moduli space of curves.
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09.05.2018 | Riccardo Moschetti (Uni Stavanger) | |

Title: Equivalence of K3 surfaces from Verra threefolds |
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Abstract: Two varieties are said to be L-equivalent if the difference of their classes in the Grothendieck ring is annihilated by a certain power of the class of the affine line. It is conjectured that two simply connected varieties with equivalent derived category are L-equivalent. I will talk about an evidence for this conjecture arising in the context of Verra threefolds, (2,2) divisors of P^2 \times P^2. This is a joint work with Grzegorz and Michał Kapustka.
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16.05.2018 | kein Seminar | |

23.05.2018 | Alexander Polishchuk (Uni Oregon) | |

Title: Moduli spaces of curves with nonspecial divisors |
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Abstract: In this talk I will discuss the moduli spaces of pointed curves with possibly non-nodal singularities such that the marked points form a nonspecial ample divisor. I will show that such curves have natural projective embeddings, with a canonical choice of homogeneous coordinates up to rescaling. Using Groebner bases technique this leads to the identification of the moduli stack with some global quotient by a torus action. Looking at the corresponding GIT quotients one gets birational projective models of M_{g,n}, some of which can be explicitly described.
As another application, I will construct a birational morphism contracting the Weierstrass divisor in M_{g,1} to a point.
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30.05.2018 | Raju Krishnamoorty (FU Berlin) | |

Title: Analogs of the Hasse Invariant |
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Abstract: We'll use formal properties of correspondences without a core to sketch conceptual (i.e. non-computational) proofs of statements like the following.
1. Any two supersingular elliptic curves over F_{p} are related by an l-primary isogeny for any l ≠ p.
2. A Hecke correspondence of compactified modular curves is always ramified at at least one cusp.
3. There is no canonical lift of supersingular points on a (projective) Shimura curve. (In particular, this provides yet another conceptual reason why there is not a canonical lift of supersingular elliptic curves.)
To do this, we'll introduce the concepts of invariant line bundles and of invariant sections on a correspondence without a core. Then (1), (2), and (3) will be implied by the following:
Theorem 1: Let X<-Z->Y be a correspondence of curves without a core over a field k. There is at most one etale clump.
Theorem 2. Let X<-Z->Y be an etale correspondence of curves without a core over a field of characteristic 0. Then there are no clumps.
We'll end with several open questions.
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06.06.2018 | kein Seminar | |

13.06.2018 | 13:00 - 14:15 Olivier Benoist (ENS Paris) | |

Title: A real period-index theorem |
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Abstract: De Jong has proven that the period and the index of a class in the Brauer group of the function field of a complex surface coincide.
We prove the same statement for classes in the Brauer group of the function field of a real surface that are trivial in restriction to the real points of the surface.
As a consequence, we show that the u-invariant of the function field of a real surface is equal to 4.
In this talk, after explaining and motivating these statements, we will sketch the proof, that relies on Hodge theory.
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14:45 - 16:00 Wushi Goldring (Stockholm University) | ||

Title: The analogy between Mumford-Tate domains over C and G-Zips mod p |
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Abstract: I will talk about our program that aims to connect (A) Automorphic Algebraicity, (B) G-Zip Geometricity, and (C) Griffiths-Schmid Algebraicity. It was initiated jointly with J.-S. Koksivirta and
developed further together with B. Stroh and Y. Brunebarbe. In a
nutshell, the theme of the program may be summed up as
"characteristic-shifting between automorphic spaces".
The focus of the talk will be to illustrate the analogy between stacks
of G-Zips in characteristic p and Mumford-Tate domains over C. I will
explain how the pursuit of the analogy leads to new results and
conjectures, both in characteristic p and over C. Two examples
include: (1) Proof and generalization of a conjecture of F. Diamond on
mod p Hilbert modular forms, (2) Extension of some results of
Griffiths-Schmid on positivity of automorphic bundles.
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20.06.2018 | Herbert Lange (FAU Erlangen-Nürnberg) | |

Title: Etale double covers of cyclic p-gonal covers |
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Abstract: In the first part of the talk I will compute the Galois group of the Galois closure
of the composition of an etale double cover and a cyclic p-gonal cover, where p is a prime.
The main result is a relation between the Prym variety of the double cover and the
Jacobian of a certain subcover. For p=3 this gives a new proof of the trigonal construction.
For higher p this can be considered as a generalization of it.
This is joint work with A. Carocca and R. Rodriguez.
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27.06.2018 | Einstein Lectures in Algebraic Geometry | |

13:15 - 14:15 Scott Mullane (Uni Georgia) | ||

Title: Extremal and rigid effective cycles in ℳ_{g,n} |
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Abstract: A number of questions about geometry and dynamics on Riemann surfaces reduce to studying the strata of abelian differentials with prescribed number and multiplicities of zeros and poles. In this talk, we will focus on the divisorial strata closures that form special codimension-one subvarieties in ℳ_{g,n}. For genus g>1 curves with n>g marked points, we show that infinitely many of these divisors form extremal rays of the cone of effective divisors. Hence these effective cones are not rational polyhedral. We'll then discuss how this result informs the cones of higher codimension effective cycles.
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14:45 - 15:45 Hsian-Hua Tseng (Ohio State University) | ||

Title: The descendant Hilb/Sym correspondence for the plane |
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Abstract: Let S be a nonsingular surface. A version of the crepant resolution conjecture predicts that the descendant Gromov-Witten theory of Hilb^{n}(S), the Hilbert scheme of n points on S, is equivalent to the descendant Gromov-Witten theory of Sym^{n}(S), the n-fold symmetric product of S. In this talk we discuss how this works when S is ℂ^{2}. We explicitly identify a symplectic transformation equating the two descendant Gromov-Witten theories. We also establish a relationship between this symplectic transformation and the Fourier-Mukai transformation which identifies the (torus-equivariant) K-groups of Hilb^{n}(ℂ^{2}) and Sym^{n}(ℂ^{2}). This is based on joint work with R. Pandharipande.
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04.07.2018 | Maksym Fedorchuk (Boston College) | |

Title: Geometry of the associated form morphism |
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Abstract: The associated form morphism is an algebraically constructed morphism from the space of smooth degree d hypersurfaces in a n-dimensional projective space to the space of (GIT semistable) degree (n+1)(d-2) hypersurfaces in the dual space. I will discuss a surprising geometric property of this morphism: The fact that it descends to give a locally closed immersion on the levels of GIT quotients, and that in the resulting new compactification of the GIT moduli space of smooth hypersurfaces the discriminant divisor is often contracted (sometimes to a point). This is a joint (forthcoming) work with Alexander Isaev. If time permits, I will describe a few explicit examples of this morphism on moduli spaces of points on the projective line, and moduli spaces of plane curves.
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11.07.2018 | Joe Harris (Harvard University) | |

Title: The Maximal Rank Theorem |
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Abstract: The Brill-Noether theorem establishes a fundamental link
between the classical notion of a curve in projective space, given as
the zero locus of polynomials, and the relatively modern notion of an
abstract curve. Specifically, it tell us when and how a general abstract
curve can be embedded in P^{r}.
But that’s just the opening line of the story: having embedded our
abstract curve in projective space, we can ask about the geometry and
algebra of the image. In particular, we ask what sort of polynomial
equations define the image — what their degrees are, and how many of
them there are. The "Maximal Rank Conjecture", recently proved by Eric
Larson, gives an answer to this question. In this talk, I’ll describe
the ideas leading up to this theorem, give an overview of the proof, and
discuss the questions that follow.
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18.07.2018 | kein Seminar | |

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