Math 245A: Resolution of singularities (Stanford, Autumn Quarter 2023)

Meets every Tuesday & Thursday, 1:30PM -- 2:50PM

Venue: 50-52H


PART I: Characteristic 0

Week 1 (September 26 & 28): Outline of goals.

Week 2 (October 3 & 5): Overview of previous strategies in Hironaka, Bierstone--Milman, Encinas--Villamayor, with a view toward recent methods.

Week 3 (October 10 & 12): Hypersurfaces of maximal contact: existence in characteristic zero and formal uniqueness. Coefficient ideal.

Week 4 (October 17 & 19): Zariski--Riemann space, Rees algebras, Hironaka's idealistic exponents, weighted blow-ups, deformation to (weighted) normal cone.

Week 5 (October 24 & 26): Invariant associated to an embedded singularity. Unique admissible blow-up center with maximal invariant. Resolution of singularities via smooth weighted blow-ups. Destackification.

Week 6 (October 31 & November 2): Log structures and their role in log resolution of singularities. Log smoothness.

Week 7 (November 7 & 9): Toric weighted blow-ups; more generally, toroidal weighted blow-ups. Log resolution of singularities via toroidal weighted blow-ups. Resolution of toroidal singularities.

Week 8 (November 14 & 16): Cox's construction; more generally, Satriano's construction of canonical Artin stacks over log smooth schemes. Multi-weighted blow-ups on affine spaces. Log resolution of singularities via multi-weighted blow-ups. Reduction of stabilizers.

Supplementary reading

More to be added...

PART II: Arbitrary characteristic

Week 9 (November 21 & 23): Thanksgiving recess. NO CLASS.

Week 10 (November 28 & 30): Regularity and the Hilbert--Samuel function. Normal flatness. Non-existence of maximal contact in characteristic p > 0. Directrix and ridge.

Week 11 (December 5 & 7): Some results of Bennett, Hironaka, Mizutani. Hironaka's characteristic polyhedron. Resolution in dimension 2.

Main references