Inner geometry of complex surfaces: a valuative approach

Silva, André Belotto da; Fantini, Lorenzo; Pichon, Anne

doi:10.2140/gt.2022.26.163

Cited by 6 publications

(29 citation statements)

References 35 publications

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“…By definition, a vertex w of Γ σ is the root vertex (respectively, a Δ-node) of Γ σ if and only if −1 (w) contains an L-node (respectively a P-node) of (X, 0). To prove part (i) of the proposition it is then sufficient to establish the following claim: a vertex w of Γ σ that is not On Lipschitz normally embedded complex surface germs (v) The inner rates of the vertices of Γ σ Δ are determined by those of the vertices of Γ π because inner rates commute with thanks to [BFP22,Lemma 3.2].…”

Section: Discriminant Curvesmentioning

confidence: 99%

“…By Corollary 4.6, there exist a refinement of and a sequence of point blowups such that induces a morphism of graphs . In particular, because respects inner rates (see again [BFP22, Lemma 3.2]), all edges of that are incoming at are sent to the unique edge of that is incoming at (its uniqueness can for example be seen as a consequence of Proposition 5.1). The lemma follows now by applying the formula of [BFP22, Lemma 4.18], observing that the local degree along every edge adjacent to equals 1 by Lemma 4.1(ii) and that, even if further point blowups may be needed to pass from to a resolution adapted to , no new edge can be incoming at .…”

Section: End Of the Proof Of Theorem 11mentioning

confidence: 99%

“…Observe that, because the exceptional component of the blowup of an intersection point between the two components associated with and has multiplicity , and , the metric on is compatible with subdividing the edges of the graph by blowing up at double points of , and thus induces a metric on . The reader should be warned that this metric on is not the same as that defined in [BFP22, § 2.1], albeit it is strictly related to the latter and was already briefly used in Lemma 5.5 of [BFP22].…”

Section: Inner Rates On Lne Surface Germsmentioning

confidence: 99%

“…Denote by the unique -node of , that is, the divisorial point associated with the blowup of at . For every divisorial point of the inner rate of is by [BFP22, Lemma 5.5] (or, in a more elementary way, by a simple computation using Lemma 3.6 of [BFP22]). As the inner rates on and commute with the map (see [BFP22, Lemma 3.2]), we need to show that .…”

Section: Inner Rates On Lne Surface Germsmentioning

confidence: 99%

“…Although the first two parts of the theorem are quite elementary, the remaining parts rely heavily on a careful study of generic projections of LNE surfaces (see Lemma 4.1), building on results from [NPP20a]. Parts (iii) and (iv) also depend on the study of inner rates of [BFP22], and, in particular, on the so-called Laplacian formula of [BFP22].…”

Section: Introductionmentioning

confidence: 99%

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On Lipschitz normally embedded complex surface germs

Silva¹,

Fantini²,

Pichon³

2022

Compositio Math.

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We undertake a systematic study of Lipschitz normally embedded normal complex surface germs. We prove, in particular, that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz normally embedded surface singularity.

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Section: Discriminant Curvesmentioning

confidence: 99%

Section: End Of the Proof Of Theorem 11mentioning

confidence: 99%

Section: Inner Rates On Lne Surface Germsmentioning

confidence: 99%

Section: Inner Rates On Lne Surface Germsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Lipschitz normally embedded complex surface germs

Silva¹,

Fantini²,

Pichon³

2022

Compositio Math.

Self Cite

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On Lipschitz Normally Embedded Singularities

Fantini,

Pichon

2023

Handbook of Geometry and Topology of Singularities IV

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Inner rates of finite morphisms

Cherik

2024

Math. Z.

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Let (X, 0) be a complex analytic surface germ embedded in $$({\mathbb {C}}^n,0)$$ ( C n , 0 ) with an isolated singularity and $$\Phi =(g,f):(X,0) \longrightarrow ({\mathbb {C}}^2,0)$$ Φ = ( g , f ) : ( X , 0 ) ⟶ ( C 2 , 0 ) be a finite morphism. We define a family of analytic invariants of the morphism $$\Phi $$ Φ , called inner rates of $$\Phi $$ Φ . By means of the inner rates we study the polar curve associated with the morphism $$\Phi $$ Φ when fixing the topological data of the curve $$(gf)^{-1}(0)$$ ( g f ) - 1 ( 0 ) and the surface germ (X, 0), allowing to address a problem called polar exploration. We also use the inner rates to study the geometry of the Milnor fibers of a non constant holomorphic function $$f:(X,0) \longrightarrow ({\mathbb {C}},0)$$ f : ( X , 0 ) ⟶ ( C , 0 ) . The main result is a formula which involves the inner rates and the polar curve alongside topological invariants of the surface germ (X, 0) and of the curve $$(gf)^{-1}(0)$$ ( g f ) - 1 ( 0 ) .

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Inner geometry of complex surfaces: a valuative approach

Cited by 6 publications

References 35 publications

On Lipschitz normally embedded complex surface germs

On Lipschitz normally embedded complex surface germs

On Lipschitz Normally Embedded Singularities

Inner rates of finite morphisms

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