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Displaying similar documents to “Varieties of minimal rational tangents of codimension 1”

Explicit birational geometry of threefolds of general type, I

Jungkai A. Chen, Meng Chen (2010)

Annales scientifiques de l'École Normale Supérieure

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Let V be a complex nonsingular projective 3-fold of general type. We prove P 12 ( V ) : = dim H 0 ( V , 12 K V ) > 0 and P m 0 ( V ) > 1 for some positive integer m 0 24 . A direct consequence is the birationality of the pluricanonical map ϕ m for all m 126 . Besides, the canonical volume Vol ( V ) has a universal lower bound ν ( 3 ) 1 63 · 126 2 .

The local lifting problem for actions of finite groups on curves

Ted Chinburg, Robert Guralnick, David Harbater (2011)

Annales scientifiques de l'École Normale Supérieure

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Let k be an algebraically closed field of characteristic p > 0 . We study obstructions to lifting to characteristic 0 the faithful continuous action φ of a finite group G on k [ [ t ] ] . To each such  φ a theorem of Katz and Gabber associates an action of G on a smooth projective curve Y over k . We say that the KGB obstruction of φ vanishes if G acts on a smooth projective curve X in characteristic  0 in such a way that X / H and Y / H have the same genus for all subgroups H G . We determine for which G the KGB...

On the Picard number of divisors in Fano manifolds

Cinzia Casagrande (2012)

Annales scientifiques de l'École Normale Supérieure

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Let  X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in  X . We consider the image 𝒩 1 ( D , X ) of  𝒩 1 ( D ) in  𝒩 1 ( X ) under the natural push-forward of 1 -cycles. We show that ρ X - ρ D codim 𝒩 1 ( D , X ) 8 . Moreover if codim 𝒩 1 ( D , X ) 3 , then either X S × T where S is a Del Pezzo surface, or codim 𝒩 1 ( D , X ) = 3 and X has a fibration in Del Pezzo surfaces onto a Fano manifold T such that ρ X - ρ T = 4 .

Brill–Noether loci for divisors on irregular varieties

Margarida Mendes Lopes, Rita Pardini, Pietro Pirola (2014)

Journal of the European Mathematical Society

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We take up the study of the Brill-Noether loci W r ( L , X ) : = { η Pic 0 ( X ) | h 0 ( L η ) r + 1 } , where X is a smooth projective variety of dimension > 1 , L Pic ( X ) , and r 0 is an integer. By studying the infinitesimal structure of these loci and the Petri map (defined in analogy with the case of curves), we obtain lower bounds for h 0 ( K D ) , where D is a divisor that moves linearly on a smooth projective variety X of maximal Albanese dimension. In this way we sharpen the results of [Xi] and we generalize them to dimension > 2 . In the 2 -dimensional case...

On the birational gonalities of smooth curves

E. Ballico (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let C be a smooth curve of genus g . For each positive integer r the birational r -gonality s r ( C ) of C is the minimal integer t such that there is L Pic t ( C ) with h 0 ( C , L ) = r + 1 . Fix an integer r 3 . In this paper we prove the existence of an integer g r such that for every integer g g r there is a smooth curve C of genus g with s r + 1 ( C ) / ( r + 1 ) > s r ( C ) / r , i.e. in the sequence of all birational gonalities of C at least one of the slope inequalities fails.

On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let h i ( k , n ) be the i -th ordered configuration space of all distinct points H 1 , ... , H h in the Grassmannian G r ( k , n ) of k -dimensional subspaces of n , whose sum is a subspace of dimension i . We prove that h i ( k , n ) is (when non empty) a complex submanifold of G r ( k , n ) h of dimension i ( n - i ) + h k ( i - k ) and its fundamental group is trivial if i = m i n ( n , h k ) , h k n and n > 2 and equal to the braid group of the sphere P 1 if n = 2 . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k = n - 1 .

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

On sums and products in a field

Guang-Liang Zhou, Zhi-Wei Sun (2022)

Czechoslovak Mathematical Journal

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We study sums and products in a field. Let F be a field with ch ( F ) 2 , where ch ( F ) is the characteristic of F . For any integer k 4 , we show that any x F can be written as a 1 + + a k with a 1 , , a k F and a 1 a k = 1 , and that for any α F { 0 } we can write every x F as a 1 a k with a 1 , , a k F and a 1 + + a k = α . We also prove that for any x F and k { 2 , 3 , } there are a 1 , , a 2 k F such that a 1 + + a 2 k = x = a 1 a 2 k .

Purity of level m stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

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Let k be a field of characteristic p &gt; 0 . Let D m be a BT m over k (i.e., an m -truncated Barsotti–Tate group over k ). Let S be a k -scheme and let X be a BT m over S . Let S D m ( X ) be the subscheme of S which describes the locus where X is locally for the fppf topology isomorphic to D m . If p 5 , we show that S D m ( X ) is pure in S , i.e. the immersion S D m ( X ) S is affine. For p { 2 , 3 } , we prove purity if D m satisfies a certain technical property depending only on its p -torsion D m [ p ] . For p 5 , we apply the developed techniques to show that...

An effective proof of the hyperelliptic Shafarevich conjecture

Rafael von Känel (2014)

Journal de Théorie des Nombres de Bordeaux

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Let C be a hyperelliptic curve of genus g 1 over a number field K with good reduction outside a finite set of places S of K . We prove that C has a Weierstrass model over the ring of integers of K with height effectively bounded only in terms of g , S and K . In particular, we obtain that for any given number field K , finite set of places S of K and integer g 1 one can in principle determine the set of K -isomorphism classes of hyperelliptic curves over K of genus g with good reduction outside...