Holomorphic self-maps of singular rational surfaces

Favre, Charles

doi:10.5565/publmat_54210_06

Cited by 21 publications

(30 citation statements)

References 59 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If ∆ = 0 and X is a surface this follows from a theorem of Wahl [Wah90], cf. also Favre [Fav10]. More generally if ∆ = 0 and X has at most isolated singularities, we can apply [BdFF12, Thm.B] or [Ful11,Cor.].…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Singularities of varieties admitting an endomorphism

Broustet

Höring

2014

Math. Ann.

View full text Add to dashboard Cite

Abstract. Let X be a normal variety such that K X is Q-Cartier, and let f : X → X be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that are not log-canonical and the dynamics of the endomorphism f . As a consequence we prove that if X is projective and f polarised, then X has at most log-canonical singularities.

show abstract

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Singularities of varieties admitting an endomorphism

Broustet

Höring

2014

Math. Ann.

View full text Add to dashboard Cite

show abstract

“…In this section we recall the notion of angular distance ρ of prime divisorial valuations (see Definition 1. 13), introduced in a greater generality by Gignac and the last author in [20] and by the first three authors in a slightly different form in [19] for the restricted class of arborescent singularities. The definition uses the bracket of Definition 1.6.…”

Section: The Angular Distancementioning

confidence: 99%

“…Any semivaluation on X induces a dual divisor on X π , according to the next proposition (see [13,Page 400]…”

Section: B-divisors On Normal Surface Singularitiesmentioning

confidence: 99%

“…Tangent directions need to be thought as branches at a point z of W , and in some way as infinitesimal objects (hence the name tangent direction). For this reason we distinguish an element − → v ∈ T z W from the set The structure of the space of normalized semivaluations V X associated to a normal surface singularity X has been investigated from a viewpoint similar to that of the present paper by Favre [13], and by Gignac and the last-named author in [20]. It has also been investigated from somewhat different perspectives by Fantini [11,12], Thuillier [48] and de Felipe [8].…”

Section: R-trees and Graphs Of R-treesmentioning

confidence: 99%

See 1 more Smart Citation

Ultrametric properties for valuation spaces of normal surface singularities

Barroso

Pérez

Popescu-Pampu

et al. 2019

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Let L be a fixed branch -that is, an irreducible germ of curve -on a normal surface singularity X. If A, B are two other branches, define uL(A, B) :where A · B denotes the intersection number of A and B. Call X arborescent if all the dual graphs of its good resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P loski by proving that whenever X is arborescent, the function uL is an ultrametric on the set of branches on X different from L. In the present paper we prove that, conversely, if uL is an ultrametric, then X is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on X, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which uL is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing L to be an arbitrary semivaluation on X and by defining uL on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if X is arborescent, and without any restriction on X we exhibit special subspaces of the space of semivaluations in restriction to which uL is still an ultrametric.

show abstract

“…Main ingredients of the proofs. The results of Favre [9], Nakayama [25] and Wahl [28] are very inspiring about the restriction of the singularity type of a normal surface imposed by the existence of an endomorphism of degree > 1 on the surface. For the proof of our results, the basic ingredients are: a log canonical singularity criterion (Proposition 2.1), the inversion of adjunction in Kawakita [14], a rational connectedness criterion of Qi Zhang [31] and its generalization in Hacon-McKernan [12], the characterization in Mori [23] on hypersurfaces in weighted projective spaces, and the equivariant Minimal Model Program in our early paper [29].…”

Section: Introductionmentioning

confidence: 99%

Invariant hypersurfaces of endomorphisms of projective varieties

Zhang

2014

Advances in Mathematics

View full text Add to dashboard Cite

We consider surjective endomorphisms f of degree > 1 on projective manifolds X of Picard number one and their f −1 -stable hypersurfaces V , and show that V is rationally chain connected. Also given is an optimal upper bound for the number of f −1 -stable prime divisors on (not necessarily smooth) projective varieties.The conditions (1) and (2) in Theorem 1.1 are satisfied if X is smooth with Picard number ρ(X) = 1. The ampleness of V i in Theorem 1.1 above and the related result

show abstract

Holomorphic self-maps of singular rational surfaces

Cited by 21 publications

References 59 publications

Singularities of varieties admitting an endomorphism

Singularities of varieties admitting an endomorphism

Ultrametric properties for valuation spaces of normal surface singularities

Invariant hypersurfaces of endomorphisms of projective varieties

Contact Info

Product

Resources

About