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Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.

5 votes

Q-factorial and rational singularities on surfaces

About the converse: one does need all the assumptions Karl mentioned in his answer. There are $2$-dim. complete local rings which are UFD but does not have rational singularity. One such example (due …
Hailong Dao's user avatar
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12 votes
Accepted

Two conjectures by Gabber on Brauer and Picard groups

Happy news: there have been some progress over the last 10 years. The hypersurface case of Conjecture 2 was proved in H. Dao, Picard groups of punctured spectra of dimension three local hypersurface …
Hailong Dao's user avatar
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21 votes
1 answer
2k views

Two conjectures by Gabber on Brauer and Picard groups

In a paper I need to make reference to two conjectures by Gabber, from Ofer Gabber, On purity for the Brauer group, in: Arithmetic Algebraic Geometry, MFO Report No. 37/2004, doi:10.14760/OWR-2004-37 …
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5 votes

Have people successfully worked with the full ring of differential operators in characterist...

Dear Matt: The people who are actively working with this whom I know are Genady Lyubeznik and Manuel Blickle. The key point seems to be that certain $R[F]$-modules become simpler when viewed as $D_R$- …
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