Hyperbolicity of Projective Hypersurfaces

Diverio, Simone; Rousseau, Erwan

doi:10.1007/978-3-319-32315-2

Cited by 14 publications

(12 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This bundle of graded algebras

E_{k}^{n} = \oplus_{m} E_{k, m}^{n}

has been an important object of study for a long time. The invariant jet differentials play a crucial role in the strategy developed by Green, Griffiths, Bloch, Ahlfors, Demailly, Siu and others to prove Kobayashi's 1970 hyperbolicity conjecture .…”

Section: Jet Differentials and Generalised Demailly–semple Jet Bundlesmentioning

confidence: 99%

“…Our first motivation for considering linear actions of groups of the form

\overset{U}{true}

in this article and in came from the study of jet differentials. The groups

G_{k}

k

‐jets of holomorphic reparametrisations of

(C, 0)

(and more generally the groups

G_{k, p}

k

‐jets of holomorphic reparametrisations of

(C^{p}, 0)

for

p ⩾ 1

) play an important role in the strategy of Demailly, Siu and others towards the Green–Griffiths conjecture on entire holomorphic curves in hypersurfaces of large degree in projective spaces. Here

G_{k}

is a non‐reductive complex linear algebraic group which is a semidirect product

{double-struckG}_{k} = {double-struckU}_{k} ⋊ C^{*}

of its unipotent radical

U_{k}

{double-struckC}^{*}

acting with weights

1, 2, 3, \dots, k

on the Lie algebra of

U_{k}

, while if

p > 1

then

{double-struckG}_{k, p} = {double-struckU}_{k, p} ⋊ G L false(p; double-struckC false)

where all the weights of the central one‐parameter subgroup

{double-struckC}^{*}

G L (p; C)

on the Lie algebra of the unipotent radical

…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Geometric invariant theory for graded unipotent groups and applications

Bérczi

Doran

Hawes

et al. 2018

Journal of Topology

121

View full text Add to dashboard Cite

Let U be a graded unipotent group over the complex numbers, in the sense that it has an extensionÛ by the multiplicative group such that the action of the multiplicative group by conjugation on the Lie algebra of U has all its weights strictly positive. Given any action of U on a projective variety X extending to an action ofÛ which is linear with respect to an ample line bundle on X, then provided that one is willing to replace the line bundle with a tensor power and to twist the linearisation of the action ofÛ by a suitable (rational) character, and provided an additional condition is satisfied which is the analogue of the condition in classical geometric invariant theory (GIT) that there should be no strictly semistable points for the action, we show that theÛ -invariants form a finitely generated graded algebra; moreover, the natural morphism from the semistable subset of X to the enveloping quotient is surjective and expresses the enveloping quotient as a geometric quotient of the semistable subset. Applying this result with X replaced by its product with the projective line gives us a projective variety which is a geometric quotient byÛ of an invariant open subset of the product of X with the affine line and contains as an open subset a geometric quotient of a U -invariant open subset of X by the action of U . Furthermore, these open subsets of X and its product with the affine line can be described using criteria similar to the Hilbert-Mumford criteria in classical GIT.

show abstract

“…This bundle of graded algebras

E_{k}^{n} = \oplus_{m} E_{k, m}^{n}

Section: Jet Differentials and Generalised Demailly–semple Jet Bundlesmentioning

confidence: 99%

“…Our first motivation for considering linear actions of groups of the form

\overset{U}{true}

in this article and in came from the study of jet differentials. The groups

G_{k}

k

‐jets of holomorphic reparametrisations of

(C, 0)

(and more generally the groups

G_{k, p}

k

‐jets of holomorphic reparametrisations of

(C^{p}, 0)

for

p ⩾ 1

) play an important role in the strategy of Demailly, Siu and others towards the Green–Griffiths conjecture on entire holomorphic curves in hypersurfaces of large degree in projective spaces. Here

G_{k}

is a non‐reductive complex linear algebraic group which is a semidirect product

{double-struckG}_{k} = {double-struckU}_{k} ⋊ C^{*}

of its unipotent radical

U_{k}

{double-struckC}^{*}

acting with weights

1, 2, 3, \dots, k

on the Lie algebra of

U_{k}

, while if

p > 1

then

{double-struckG}_{k, p} = {double-struckU}_{k, p} ⋊ G L false(p; double-struckC false)

where all the weights of the central one‐parameter subgroup

{double-struckC}^{*}

G L (p; C)

on the Lie algebra of the unipotent radical

…”

Section: Introductionmentioning

confidence: 99%

Geometric invariant theory for graded unipotent groups and applications

Bérczi

Doran

Hawes

et al. 2018

Journal of Topology

121

View full text Add to dashboard Cite

show abstract

“…The strategy of the proof [15,17,18,24,49]. Proving the algebraic degeneracy of holomorphic curves on X means finding a non zero polynomial P on X such that all entire curves f : C → X satisfy P( f (C)) = 0.…”

Section: Highlights In the History Of The Green-griffiths Conjecturementioning

confidence: 99%

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

Bérczi¹

2012

EMS Series of Congress Reports

View full text Add to dashboard Cite

show abstract

“…Two important sources of interest in bounding genera are: the Clemens Conjecture on count of rational curves in Calabi-Yau varieties ( [27]), and the celebrated Kobayashi Conjecture on hyperbolicity of general hypersurfaces in P n of sufficiently large degree ( [59]). Recall ( [43], [59]) that the Kobayashi hyperbolicity of a variety X implies the algebraic hyperbolicity, and in particular, absence of rational and elliptic curves in X. A part concerning the Clemens Conjecture started with the following theorem.…”

Section: Genera Of Subvarieties: a Surveymentioning

confidence: 99%

A remark on the intersection of plane curves

Ciliberto

Flamini²,

Zaidenberg

2019

Functional Analysis and Geometry

View full text Add to dashboard Cite

Let D be a very general curve of degree d = 2ℓ − ε in P 2 , with ε ∈ {0, 1}. Let Γ ⊂ P 2 be an integral curve of geometric genus g and degree m, Γ = D, and let ν : C → Γ be the normalization. Let δ be the degree of the reduction modulo 2 of the divisor ν * (D) of C (see § 2.1). In this paper we prove the inequality 4g + δ m(d − 8 + 2ε) + 5. We compare this with similar inequalities due to Geng Xu ([88, 89]) and Xi Chen ([17, 18]). Besides, we provide a brief account on genera of subvarieties in projective hypersurfaces.

show abstract

Hyperbolicity of Projective Hypersurfaces

Cited by 14 publications

References 29 publications

Geometric invariant theory for graded unipotent groups and applications

Geometric invariant theory for graded unipotent groups and applications

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

A remark on the intersection of plane curves

Contact Info

Product

Resources

About