Moret-Bailly families   and non-liftable schemes

Rössler, Damian; Schröer, Stefan

doi:10.14231/ag-2022-004

Cited by 5 publications

(3 citation statements)

References 12 publications

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“…Moreover, the term on the right also vanishes because G acts via the sign involution on the cohomology group, according ( [27], proof of Proposition 2.3). This establishes the claim.…”

Section: The Picard Scheme Of the Quotientmentioning

confidence: 92%

“…A similar situation with N * = (Z/2Z) k and N = µ p arise if there is a point of order two on Pic B/k . In both cases the discussion in [27], beginning of Section 2 shows that A has the structure of an abelian variety so that the projection A → A is a homomorphism, and we get an inclusion N ⊂ A . The composition A → B is the quotient by the group scheme N {±1}.…”

Section: The Picard Scheme Of the Quotientmentioning

confidence: 97%

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Sign involutions on para-abelian varieties

Bergqvist¹,

Dang²,

Schröer³

2023

Preprint

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We study the so-called sign involutions on twisted forms of abelian varieties, and show that such a sign involution exists if and only if the class in the Weil-Châtelet group is annihilated by two. If these equivalent conditions hold, we prove that the Picard scheme of the quotient is étale and contains no points of finite order. In dimension one, such quotients are Brauer-Severi curves, and we analyze the ensuing embeddings of the genus-one curve into twisted forms of Hirzebruch surfaces and weighted projective spaces. ContentsIntroduction 1 1. The scheme of sign involutions 3 2. The Picard scheme of the quotient 6 3. Morphisms to Brauer-Severi curves 9 References 13

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“…Moreover, the term on the right also vanishes because G acts via the sign involution on the cohomology group, according ( [27], proof of Proposition 2.3). This establishes the claim.…”

Section: The Picard Scheme Of the Quotientmentioning

confidence: 92%

Section: The Picard Scheme Of the Quotientmentioning

confidence: 97%

Sign involutions on para-abelian varieties

Bergqvist¹,

Dang²,

Schröer³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Over the field extension k = k(t 1/p ), the section given by z → (z, t 1/p z) defines the desired splitting U ⊗k (G a ×α p )⊗k . Furthermore, there is a supersingular elliptic curve N over k (use for example [42], Lemma 3.1), which indeed contains a copy of α p .…”

Section: The Case Of Algebraic Groupsmentioning

confidence: 99%

Albanese maps for open algebraic spaces

Schröer¹

2022

Preprint

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We show that for each algebraic space that is separated and of finite type over a field, and whose affinization is connected and reduced, there is a universal morphism to a para-abelian variety. The latter are schemes that acquire the structure of an abelian variety after some ground field extension. This generalizes classical results of Serre on universal morphisms from algebraic varieties to abelian varieties. Our proof relies on corresponding facts for the proper case, together with the theory of Macaulayfication, removal of singularities by alterations, pseudo-rational singularities, ind-objects, and Bockstein maps. It turns out that the formation of the Albanese variety commutes with base-change up to universal homeomorphisms. We also give a detailed analysis of Albanese maps for algebraic curves and algebraic groups, with special emphasis on imperfect ground fields.Contents 21 8. The case of algebraic groups 25 References 29

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Para-Abelian Varieties and Albanese Maps

Laurent,

Schröer

2023

Bull Braz Math Soc, New Series

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We construct for every proper algebraic space over a ground field an Albanese map to a para-abelian variety, which is unique up to unique isomorphism. This holds in the absence of rational points or ample sheaves, and also for reducible or non-reduced spaces, under the mere assumption that the structure morphism is in Stein factorization. It also works under suitable assumptions in families. In fact the treatment of the relative setting is crucial, even to understand the situation over ground fields. This also ensures that Albanese maps are equivariant with respect to actions of group schemes. Our approach depends on the notion of families of para-abelian varieties, where each geometric fiber admits the structure of an abelian variety, and representability of tau-parts in relative Picard groups, together with structure results on algebraic groups.

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Moret-Bailly families and non-liftable schemes

Cited by 5 publications

References 12 publications

Sign involutions on para-abelian varieties

Sign involutions on para-abelian varieties

Albanese maps for open algebraic spaces

Para-Abelian Varieties and Albanese Maps

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