Severi inequality for varieties of maximal Albanese dimension

Zhang, Tong

doi:10.1007/s00208-014-1025-7

Cited by 27 publications

(28 citation statements)

References 16 publications

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“…Finally, [CF18, Table 3] contains three families of surfaces with K 2 X = 2 whose minimal models have K 2 = 4χ(O) = 4, so realizing the equality of Severi inequality. Indeed the description we obtain of their Albanese morphisms is coherent with the known characterization of these surfaces in [BPS16], [Zha14].…”

Section: Proofsupporting

confidence: 81%

Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms

Pignatelli

2019

Rev Mat Complut

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We study mixed surfaces, the minimal resolution S of the singularities of a quotient (C × C)/G of the square of a curve by a finite group of automorphisms that contains elements not preserving the factors. We study them through the further quotientsAs first application we prove that if the irregularity is at least 3, then S is also minimal. The result is sharp.The main result is a complete description of the Albanese morphism of S through a determined further quotient (C ×C)/G ′ that is anétale cover of the symmetric square of a curve. In particular, if the irregularity of S is at least 2, then S has maximal Albanese dimension.We apply our result to all the semi-isogenous mixed surfaces of maximal Albanese dimension constructed by Cancian and Frapporti, relating them with the other constructions of surfaces of general type having the same invariants in the literature.

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Section: Proofsupporting

confidence: 81%

Quotients of the square of a curve by a mixed action, further quotients and Albanese morphisms

Pignatelli

2019

Rev Mat Complut

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“…The main known Clifford-Severi inequalities are the following [2] (cf. also [35] for the case L = K X ): for any nef L we have…”

Section: Introductionmentioning

confidence: 96%

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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“…Hence it suffices to assume that X is smooth. In this case, the result is just [42,Theorem 4.1]. Also see [19,Remark 1.4] for the generic vanishing theorem for singular varieties with rational singularities in characteristic 0.…”

Section: Proof Of Theorem 13mentioning

confidence: 92%

“…In this section, we will prove several relative Noether inequalities. The relative Noether inequality in [42] studies linear series on fibered varieties over curves, while the relative Noether inequalities in this section are devoted to studying fibered varieties whose fibers are curves.…”

Section: Relative Noether Inequalitiesmentioning

confidence: 99%