Oberseminar Algebra und Geometrie
Wintersemester 2016-2017:
Resolution of singularities and alterations
Organised by I. Halupczok, B. Klopsch, S. Schröer and M. Zibrowius.
All talks take place Fridays at 12:30 in 25.22.03.73.
Short description
Singularities are points where an algebraic variety, or more generally a noetherian scheme does not behave like a manifold. They occur over and over in practice, but often one would like to get rid of them. By Hironaka, an integral scheme Y of finite type over a field k of characteristic zero admits a resolution of singularities f : X →Y, which roughly means that f is surjective and generically bijective, and the scheme X has no singularities. This result is of enormous consequence, but still open in positive characteristics, let alone in arithmetic situations. De Jong proved a weaker statement in a rather general situation: Namely, there always is some f : X→Y with f surjective and generically finite, and the scheme X has no singularities. In other words, the map is an alteration rather than a modification. His main idea was to use the geometry of stable curves and their moduli. The goal of the Oberseminar is to study more elementary results on resolution of singularities, following the monograph of Cutkosky [2004], and then work through the paper of de Jong [1996].Talks and tentative dates
| 21.10.2016 | Talk 1. (Matthias Riepe)
main source: [Cutkosky 2004], Chapter 2. Introduce the notion of smoothness, regularity, and resolution of singularities. Discuss Newton's method from Section 2.1 only if time permits. |
| 28.10.2016 | Talk 2. (André Schell)
main source: [Cutkosky 2004], Chapter 3. Explain blowing-ups of the affine plane, the role of formal completions, and embedded resolutions of curves. Coordinate with next talk. |
| 04.11.2016 | Talk 3. (Benjamin Klopsch)
main source: [Cutkosky 2004], Chapter 3. Continue talk 2, and finish the proof for embedded resolutions of curves. Coordinate with previous talk. |
| 11.11.2016 | Talk 4. (Immanuel Halupczok)
main source: [Cutkosky 2004], Chapter 4. Introduce the general notion of blowing-ups of ideals, and discuss various additional conditions pertaining to resolutions of singularities. |
| 18.11.2016 | Talk 5. (Andrea Fanelli)
main source: [Cutkosky 2004], Chapter 5. Prove resolution of singularities for hypersurface singularities in dimension two over algebraically closed ground fields of characteristic zero. |
| 25.11.2016 | Talk 6. (Johannes Fischer)
main source: [de Jong 1996], Section 2. Explain the notation, conventions and terminology of the paper, skipping Subsection 2.24. |
| 02.12.2016 | Talk 7. (Kevin Langlois)
main source: [de Jong 1996], Section 3. Discuss the notion of stable pointed curves, and the corresponding moduli stack and its coarse moduli space Mg,n. Prove resolution of singularities for semistable curves, as in Proposition 3.1. |
| 09.12.2016 | Talk 8. (Sasa Novakovic)
main source: [de Jong 1996], Section 4. Explain the main result of the paper: An integral scheme X admits an alteration X1→X where the scheme X1 is regular. Coordinate with next talk. |
| 16.12.2016 | Talk 9. (Benedikt Schilson)
main source: [de Jong 1996], Section 7. Continue the previous talk if necessary, and discuss the equivariant form of the main result. Coordinate with previous talk. |
| 23.12.2016 | no session.
|
| 13.01.2017 | Talk 10. (Alexander Samokhin)
main source: [de Jong 1996], Section 5. Show that families of curves can be altered into semistable curves. |
| 20.01.2017 | talk 11. (Leif Zimmermann)
main source: [de Jong 1996], Section 6/7. Discuss variants of the main result that work over discrete valuation rings R rather than fields k. |
27.01.2017 | TBA. | 03.02.2017 | TBA. | 14.02.2017 | Program discussion for next semester. |
Literature
| [Cutkosky 2004] | Resolution of singularities. American Mathematical Society, Providence, RI. |
| [de Jong 1993] | Smoothness, semi-stability and alterations. Publ. Math., Inst. Hautes Étud. Sci. 83, 51--93. |