Generalized Clifford–Severi inequality and the volume of irregular varieties

Barja, Miguel A.

doi:10.1215/00127094-2871306

Cited by 23 publications

(43 citation statements)

References 30 publications

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“…Let us first recall the classical results, which are due mainly to Castelnuovo: see [1,Chapter III,§ 2]. Let C be a smooth curve and consider a subspace W ⊆ H 0 (C, L) of dimension r + 1 2 such that the moving part of |W | is a base point free g r d (hence deg L d).…”

Section: Continuous Castelnuovo Inequalitiesmentioning

confidence: 99%

Linear Systems on Irregular Varieties

Barja¹,

Pardini²,

Stoppino³

2019

J. Inst. Math. Jussieu

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Let X be a normal complex projective variety, T ⊆ X a subvariety, a : X → A a morphism to an abelian variety such that Pic 0 (A) injects into Pic 0 (T ) and let L be a line bundle on X.Denote by X (d) → X the connectedétale cover induced by the d-th multiplication map of A, by T (d) ⊆ X (d) the preimage of T and by L (d) the pull-back of L to X (d) . For α ∈ Pic 0 (A) general, we study the restricted linear system |L (d) ⊗ a * α| |T (d) : if for some d this gives a generically finite map ϕ (d) , we show that ϕ (d) is independent of α or d sufficiently large and divisible, and is induced by the eventual map ϕ : T → Z such that a |T factorizes through ϕ.The generic value h 0 a (X |T , L) of h 0 (X |T , L ⊗ α) is called the (restricted) continuous rank. We prove that if M is the pull back of an ample divisor of A, then x → h 0 a (X |T , L + xM ) extends to a continuous function of x ∈ R, which is differentiable except possibly at countably many points; when X = T we compute the left derivative explicitly.In the case when X and T are smooth, combining the above results we prove Clifford-Severi type inequalities, i.e., geographical bounds of the form vol X|T (L) ≥ C(m)h 0 a (X |T , L), where C(m) = O(m!).

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Section: Continuous Castelnuovo Inequalitiesmentioning

confidence: 99%

Linear Systems on Irregular Varieties

Barja¹,

Pardini²,

Stoppino³

2019

J. Inst. Math. Jussieu

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“…In particular, h 0 alb S (S, K S ) = χ(S). In [2] the following Severi-type inequalities are proved (see [2] for a much more complete statement).…”

Section: 2mentioning

confidence: 99%

“…Theorem 2.6 (Barja, [2]). Let S be a smooth surface with a generating morphism a : S → A to an abelian variety A; suppose that S is of maximal a-dimension and L is a nef divisor on S. Then (i) The following inequality holds:…”

Section: 2mentioning

confidence: 99%

“…§2.1) and then taking a limit, it is not possible to extract from it information on the case where equality holds. The breakthrough that allows us to overcome this difficulty is the work [2] by the first named author, where the Severi inequality is reinterpreted as an inequality involving the self-intersection of the nef line bundle K S and its continuous rank (cf. §2.2), and extended to the much more general situation of an arbitrary nef line bundle on a n-dimensional variety of maximal Albanese dimension (Clifford-Severi inequality).…”

Section: Introductionmentioning

confidence: 99%

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Surfaces on the Severi line

Barja

Pardini

Stoppino

2016

Journal de Mathématiques Pures et Appliquées

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Let S be a minimal complex surface of general type and of maximal Albanese dimension; by the Severi inequality one has K-S(2) >= 4 chi(O-S). We prove that the equality K-S(2) = 4 chi(O-S) holds if and only if q(S) := h(1)(Os) = 2 and the canonical model of S is a double cover of the Albanese surface branched on an ample divisor with at most negligible singularities. (C) 2015 Elsevier Masson SAS. All rights reserved.Peer ReviewedPostprint (published version

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“…In another very recent paper, Hu [23] showed that K 3 X ≥ 4 3 χ(ω X ) − 2 in this case. Also, the author has been informed by Miguel Barja that in [2,Remark 4.6], the first inequality of Theorem 1.3 is also independently proved by him using a different method.…”

Section: Introductionmentioning

confidence: 99%