Some Calabi–Yau threefolds with obstructed deformations over the Witt vectors

Schröer, Stefan

doi:10.1112/s0010437x04000545

Cited by 39 publications

(47 citation statements)

References 21 publications

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“…There seems to be a close relation between our family of K3 surfaces and the family studied in [32]. Then we have 4 = 2 2 rational points in the generic fiber and again three fibers of typeD 4 , and this leads to σ 0 = 1.…”

Section: Proposition 92 Our Supersingular K3 Surface X Has Artin Insupporting

confidence: 64%

Kummer surfaces for the self-product of the cuspidal rational curve

Schröer

2006

J. Algebraic Geom.

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The classical Kummer construction attaches a K3 surface to an abelian surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the self-product of the rational cuspidal curve, and the sign involution by suitable infinitesimal group scheme actions, we give the correct Kummer-type construction for this situation. We encounter rational double points of type D 4 and D 8 instead of type A 1 . It turns out that the resulting surfaces are supersingular K3 surfaces with Artin invariant one and two. They lie in a 1-dimensional family obtained by simultaneous resolution, which exists after purely inseparable base change. Contents STEFAN SCHRÖERmaterial on blowing up rational surface singularities along curves. We use this in Section 11, where we consider blowing ups of Weil divisors in genus-one fibrations. This is crucial in Section 12, where we view our normal K3 surfaces as lying in a family Z → S, and show that this family admits a simultaneous resolution X → S after a purely inseparable base change S → S. In the last section we show that this family is induced from a family X → P 1 .

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Section: Proposition 92 Our Supersingular K3 Surface X Has Artin Insupporting

confidence: 64%

Kummer surfaces for the self-product of the cuspidal rational curve

Schröer

2006

J. Algebraic Geom.

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“…As explained before, it is known that there are non-liftable projective Calabi-Yau threefolds in characteristics 2 and 3 (cf. [7] and [16]). It remains to prove the corresponding fact for Calabi-Yau spaces in characteristics 29, 61 and 71.…”

Section: Proof Of Theorem 11mentioning

confidence: 94%

“…After the first results of Hirokado [7] and Schröer [16], there were constructions of non-liftable Calabi-Yau spaces along these lines by Schoen [15] and independently by the first author and van Straten [4]. These constructions are based on specific models as double octics or fiber products of rational elliptic surfaces.…”

Section: Introductionmentioning

confidence: 97%

Non-liftable Calabi–Yau spaces

Cynk

Schütt

2012

Ark. Mat.

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“…Without loss of generality we may assume that A = kJx, yK, and that the action is given by x → −x and y → −y (see, for example, [13,Lemma 5.4]). The invariant ring is then kJx 2 , xy, y 2 K. Set D i = q(C i ), and let a ∈ S/{±1} be the closed point.…”

Section: Proof Let E (P) Be the Néron Model Of E (P)mentioning

confidence: 99%