Determinant via Permutations
2026TheorySergey
The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.
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170
Search results for “natural logarithm”
28 results
An example of a non-reduced Picard scheme
2026TheorySean Cotner
An example, due to Serre, of a smooth projective surface in positive characteristic whose Picard scheme fails to be reduced, worked out via the two governing dimension inequalities.
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170
Finite flat commutative group schemes embed locally into abelian schemes
2026TheorySean Cotner
A proof of Raynaud's theorem: every finite flat commutative group scheme over a locally noetherian base embeds, Zariski-locally, as a closed subgroup of an abelian scheme.
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150
The torsion component of the Picard scheme
2026OtherBogdan Zavyalov
Every finite flat commutative group scheme over a noetherian local ring is the torsion component of the Picard scheme of a smooth projective scheme with 3-dimensional fibers, built as a quotient of a complete intersection. An application: Hodge numbers that jump in a smooth projective family.
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500
A curiosity: “supersmooth” varieties
2026TheorySean Cotner
A rigidity condition on schemes, strictly stronger than smoothness, introduced with examples and a look at where it fails to be geometric — no known application, just a curiosity.
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120
Some examples of algebraic groups
2026TheorySean Cotner
Two pathological phenomena for algebraic groups over general bases — a group degenerating between the multiplicative and additive group across a DVR, and a non-affine identity component — built as centralizers in SL_n.
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180