Tetragonal curves, scrolls and 𝐾3 surfaces

Brawner, James N.

doi:10.1090/s0002-9947-97-01811-4

Cited by 14 publications

(7 citation statements)

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“…1.3] for the precise statement). It is proved in [19, §5] that one can find a pencil such that the three-dimensional rational normal scroll T ⊂ P g swept out by the span of the members of |E| in P g (which are plane cubics) is smooth (of degree g − 2) and furthermore such that (For the notion of scroll type and how to calculate it, cf., e.g., [35,5,36,19].) The possible scroll types occuring have been studied in [31, 2.11], [36, (1.7)] and [19, §9.1].…”

Section: Some Useful Resultsmentioning

confidence: 99%

“…The possible values of b 2 (and b 1 ), and the possible scroll types (e 1 , e 2 , e 3 , e 4 ), with e 1 ≥ e 2 ≥ e 3 ≥ e 4 > 0 (as T is smooth) have been investigated in [5,36,19], with some minor mistakes in the former. Recall that e 1 + e 2 + e 3 + e 4 = g − 3.…”

Section: The Case Of Clifford Index Twomentioning

confidence: 99%

“…Indeed, one easily computes h 0 (H − E) = 6, h 0 (H − 2E) = 3, h 0 (H − 3E) = 1, h 0 (H − 4E) = 0, so the resulting scroll type is (3, 2, 1, 0). A general hyperplane section of it is a rational normal scroll X of type (3, 2, 1), by [5,Thm. 2.4], whence T 0 is a cone over X.…”

Section: The Case Of Clifford Index Twomentioning

confidence: 99%

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Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections

Knutsen

2020

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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Section: Some Useful Resultsmentioning

confidence: 99%

Section: The Case Of Clifford Index Twomentioning

confidence: 99%

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Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections

Knutsen

2020

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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“…We use the following analysis of K3 surfaces of low Clifford genus, due to Reid, Brawner, and Stevens [30, section 2.11], [6,], [33, table in section 1]. Theorem 6.5.…”

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confidence: 99%

Bott vanishing for algebraic surfaces

Totaro

2020

Trans. Amer. Math. Soc.

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For William Fulton on his eightieth birthdayA smooth projective variety X over a field is said to satisfy Bott vanishing if H j (X, Ω i X ⊗ L) = 0 for all ample line bundles L, all i ≥ 0, and all j > 0. Bott proved this when X is projective space. Danilov and Steenbrink extended Bott vanishing to all smooth projective toric varieties; proofs can be found in [4,7,28,15]. What does Bott vanishing mean? It does not have a clear geometric interpretation in terms of the classification of algebraic varieties. But it is useful when it holds, as a sort of preprocessing step, since the vanishing of higher cohomology lets us compute the spaces of sections of various important vector bundles. Bott vanishing includes Kodaira vanishing as a special case (where i equals n := dim X), but it says much more.For example, any Fano variety that satisfies Bott vanishing must be rigid, since H 1 (X, T X) = H 1 (X, Ω n−1 X ⊗ K * X ) = 0 for X Fano. So Bott vanishing holds for only finitely many smooth complex Fano varieties in each dimension. Even among rigid Fano varieties, Bott vanishing fails for quadrics of dimension at least 3 and for Grassmannians other than projective space [7, section 4]. As a result, Achinger, Witaszek, and Zdanowicz asked whether a rationally connected variety that satisfies Bott vanishing must be a toric variety [1, after Theorem 4].In this paper, we exhibit several new classes of varieties that satisfy Bott vanishing. First, we answer Achinger-Witaszek-Zdanowicz's question: there are non-toric rationally connected varieties that satisfy Bott vanishing, since Bott vanishing holds for the quintic del Pezzo surface (Theorem 2.1). Over an algebraically closed field, a quintic del Pezzo surface is isomorphic to the moduli space M 0,5 of 5-pointed stable curves of genus zero. It is the only rigid del Pezzo surface that is not toric: del Pezzo surfaces of degree at least 5 are rigid, and those of degree at least 6 are toric. (The quintic del Pezzo surface also does not have a lift of the Frobenius endomorphism from Z/p to Z/p 2 , a property known to imply Bott vanishing [7], [1, Proposition 7.1.4].) In view of this example, there is a good hope of finding more Fano or rationally connected varieties that satisfy Bott vanishing.We also consider varieties that are not rationally connected, with most of the paper devoted to K3 surfaces. Bott vanishing holds for abelian varieties over any field: it reduces to Kodaira vanishing, since the tangent bundle is trivial. On the other hand, Riemann-Roch shows that Bott vanishing fails for all K3 surfaces of degree less than 20 (Theorem 3.1). But recent work of Ciliberto-Dedieu-Sernesi and Feyzbakhsh [9,14] implies: Bott vanishing holds for all K3 surfaces of degree 20 or at least 24 with Picard number 1 (Theorem 3.2). Version 2 of this paper on the arXiv gave a more elementary proof, not using Feyzbakhsh's work on Mukai's program (reconstructing a K3 surface from a curve), but here we give a short proof using her work. Surprisingly, Bott vanishing fails in degree 22.More stron...

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“…[9, Theorem 2.4] Let S = S(a 0 , a 1 , a 2 ) be a 3-dimensional rational normal scroll with index of relative balance r(S) = r. Then a general hyperplane section of S is a 2-dimensional scroll with invariants b 0 ≤ b 1 satisfying b…”

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confidence: 99%

H-instanton bundles on three-dimensional polarized projective varieties

Antonelli¹,

Malaspina²

2020

Preprint

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We propose a notion of instanton bundle (called H-instanton bundle) on any projective variety of dimension three polarized by a very ample divisor H, that naturally generalizes the ones on P 3 and on the flag threefold F (0, 1, 2). We discuss the cases of Veronese, Fano and complete intersection Calabi-Yau threefolds. Then we deal with Hinstanton bundles E on three-dimensional rational normal scrolls S(a 0 , a 1 , a 2 ). We give a monadic description of H-instanton bundles and we prove the existence of µ-stable H-instanton bundles on S(a 0 , a 1 , a 2 ) for any admissible charge k = c 2 (E)H. Then we deal in more detail with S(a, a, b) and S(a 0 , a 1 , a 2 ) with a 0 + a 1 > a 2 and even degree. Finally we describe a nice component of the moduli space of µ-stable bundles whose points represent H-instantons.

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Tetragonal curves, scrolls and 𝐾3 surfaces

Cited by 14 publications

References 4 publications

Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections

Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections

Bott vanishing for algebraic surfaces

H-instanton bundles on three-dimensional polarized projective varieties

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