Unirational surfaces on the Noether line

Liedtke, Christian; Schütt, Matthias

doi:10.2140/pjm.2009.239.343

Cited by 7 publications

(4 citation statements)

References 18 publications

(23 reference statements)

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“…The above mentioned results of Shioda [49] on the unirationality of Fermat surfaces have been generalized to Delsarte surfaces by Katsura and Shioda [51]. In [28], these have been used by Schütt and the third named author to construct unirational surfaces on the Noether line for most values of p g and in most positive characteristics p.…”

Section: 7mentioning

confidence: 96%

Deformations of rational curves in positive characteristic

Ito¹,

Ito²,

Liedtke³

2018

Preprint

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We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than (p − 1)/2 (resp. p), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth.

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Section: 7mentioning

confidence: 96%

Deformations of rational curves in positive characteristic

Ito¹,

Ito²,

Liedtke³

2018

Preprint

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“…In order to show that H 2 (X, W O X ) may not be finitely generated for surfaces on the Noether lines in arbitrary large characteristics, let us recall the following result [LS,Theorem 5.4]: Theorem 6.3 (-,Schütt). There exists an arithmetic progression P of primes of density at least 0.99999985 such that for all p ∈ P there exists an even Horikawa surface X p in characteristic p that is unirational.…”

Section: Hodge Degeneration and Crystalline Cohomologymentioning

confidence: 99%

The Canonical Map and Horikawa Surfaces in Positive Characteristic

Liedtke

2012

International Mathematics Research Notices

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We extend fundamental inequalities related to the canonical map of surfaces of general type to positive characteristic. Next, we classify surfaces on the Noether lines, i.e., even and odd Horikawa surfaces, in positive characteristic. We describe their moduli spaces and the subspaces formed by surfaces whose canonical maps are inseparable. Moreover, we compute their Betti-, deRhamand crystalline cohomology. Finally, we prove lifting to characteristic zero and show that the moduli spaces are topologically flat over the integers.

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“…In light of the few characteristic-free results on pluricanonical systems and rank 2 vector bundles on surfaces not of general type (see [E88,ShB91a]), the current focus is on surfaces of general type. Yet little is known on the classification of general type surfaces with certain positive characteristic pathologies (see [L08,LS09]). For this matter, we would like to further understand the known examples of Kodaira non-vanishing and to come up with new ones.…”

Section: Introductionmentioning

confidence: 99%

Counterexamples of Kodaira vanishing for smooth surfaces of general type in positive characteristic

Zheng

2017

Journal of Pure and Applied Algebra

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Abstract. We generalize the construction of Raynaud [R78] of smooth projective surfaces of general type in positive characteristic that violate the Kodaira vanishing theorem. This corrects an earlier paper [T10] of the same purpose. These examples are smooth surfaces fibered over a smooth curve whose direct images of the relative dualizing sheaves are not nef, and they violate Kollár's vanishing theorem. Further pathologies on these examples include the existence of non-trivial vector fields and that of non-closed global differential 1-forms.

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Unirational surfaces on the Noether line

Abstract: We show that among simply connected surfaces of general type unirationality is a common feature, even when fixing the positive characteristic or numerical invariants. To do so, we construct unirational Horikawa surfaces in abundance.

Cited by 7 publications

References 18 publications

Deformations of rational curves in positive characteristic

Deformations of rational curves in positive characteristic

The Canonical Map and Horikawa Surfaces in Positive Characteristic

Counterexamples of Kodaira vanishing for smooth surfaces of general type in positive characteristic

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