| 21.04.2010 | Gavril Farkas (HU Berlin) |
| Maps between moduli spaces of curves and Gieseker-Petri divisors | |
| Abstract: We study contractions of the moduli space of stable curves beyond the minimal model of Mg, by resolving and giving a complete enumerative description of the rational map between two moduli spaces of curves, which associates to a curve C of genus g, the Brill-Noether locus of special divisors in the case this locus is a curve. As an application we give a lower bound for the slope of the cone of moving effective divisors on Mg. | |
| Dienstag | |
| 27.04.2010 | Alessandro Verra (Universita Roma 3) |
| K3 surfaces and some moduli spaces related to curves | |
| 19.05.2010 | Paul Larsen (HU Berlin) |
| On relations in the Cox ring of the moduli space of stable pointed rational curves | |
| Abstract: The Cox ring of a projective variety encodes much of the variety's birational geometry. If the Cox ring is finitely generated, then by results of Hu and Keel, the cones of effective, nef, and moving divisors are all polyhedral. On a more basic level, finite generation implies that the Cox ring is a quotient of a polynomial ring in variables corresponding to the generators. In this talk, I will present results concerning the ideal of relations among generators of the Cox ring of the moduli space of stable pointed rational curves, \overline{M}_{0,n}. Specifically, in the polynomial ring of generators I find collections of variables with no relations among them (i.e. the corresponding subring injects into the Cox ring), and in the other direction, I find (multi) degrees that always meet the ideal of relations non-trivially. The former arise from the closely-related Losev-Manin moduli spaces, while the latter are found by studying compositions of forgetful morphisms in the Kapranov blow-up construction of \overline{M}_{0,n}. | |
| 26.05.2010 | Slawomir Cynk (Jagiellonian University, Krakow) |
| Calabi-Yau manifolds in positive characteristic that admits no lifting to characteristic 0 | |
| 02.06.2010 | Ciro Ciliberto (Universita Roma 2) |
| Gaussian maps for curves | |
| Abstract: In this talk I will recall definition and properties of gaussian maps for curves. I will then concentrate on the second gaussian map, which is related to the second fundamental form of the moduli space of curves. I will finally talk about a recent result in collaboration with A. Calabri and R. Miranda, to the effect that the second gaussian map has maximal rank for a general curve of genus g. | |
| 09.06.2010 | Sam Payne (Stanford University) |
| Tropical Brill-Noether theory | |
| Abstract:
Classical Brill-Noether theory studies the geometry of special
divisors on smooth projective curves, with special attention to the
existence of special divisors on the general curve of genus g. The
fundamental result of the theory, called the Brill-Noether Theorem and
due to Griffiths and Harris, says that a general curve of genus g has
a divisor of degree d that moves in a linear system of dimension r if
and only if g is at least (r+1)(g-d+r). The original proof uses a
degeneration to a rational curve with g nodes, plus a very subtle
transversality argument for certain associated Schubert cycles.
I will present a new and simpler proof of the Brill-Noether Theorem, using a different degeneration, to a union of smooth rational curves whose dual graph is encoded in a tropical curve with first Betti number g. The proof relies heavily on the theory of divisors on graphs pioneered by Baker and Norine, and gives an explicit criterion for a curve to be Brill-Noether general over a discretely valued field, such as Q. This is joint work with Filip Cools, Jan Draisma, and Elina Robeva. |
|
| 16.06.2010 | Stefan Kebekus (Univ. Freiburg) |
| Rationale Kurven auf algebraischen Varietäten | |
| Abstract:
Um die geometrische Struktur einer algebraischen Varietät X
zu studieren, ist es sinnvoll, Kurven zu betrachten, die in X enthalten sind.
Rationalen Kurven spielen in diesem Zusammenhang eine besondere Rolle. Dank
der bahnbrechenden Arbeiten von Shigefumi Mori ist das Studium der rationalen
Kurven auf algebraischen Varietäten inzwischen ein bewährtes und
in vielen
Bereichen unerlässliches Hilfsmittel der algebraischen Geometrie.
Anwendungen
der Theorie umfassen unter anderem Fragen zur Deformationsstarrheit
algebraischer Varietäten, Klassifikationsprobleme, oder Verallgemeinerungen
der Shafarevich Hyperbolizität.
Der Vortrag gibt einen Überblick über den Stand der Technik. Einige der weiterführenden Anwendungen werden kurz diskutiert. |
|
| 30.06.2010 | Jarod Alper (Columbia University) |
| Weakly proper moduli spaces of curves | |
| Abstract: I will present work-in-progress on the construction of modular compactifications of M_g which are expected to arise as certain log canonical models of the moduli space of stable curves of genus g. This is joint work with David Smyth and Fred van der Wyck. |