Equimultiple locus of embedded algebroid surfaces and blowing-up in characteristic zero (original) (raw)

A remark on three-sheeted algebroid surfaces whose Picard constants are five

Kodai Mathematical Journal, 1995

Let JM(R) be the family of non-constant meromorphic functions on a Riemann surface R, and P(f) be the number of values, which are not taken by Then we put P(Λ)= sup P(/), fJHOi which is called the Picard constant of R. In general P(R)^2 for every open Riemann surface R. An n-sheeted algebroid surface is a proper existence domain of an n-valued algebroid function, which is defined by the following equation:

On Plane Algebroid Curves

Two plane analytic branches are topologically equivalent if and only if they have the same multiplicity sequence. We show that having same semigroup is equivalent to having same multiplicity sequence, we calculate the semigroup from a parametrization, and we characterize semigroups for plane branches. These results are known, but the proofs are new. Furthermore we characterize multiplicity sequences of plane branches, and we prove that the associated graded ring, with respect to the values, of a plane branch is a complete intersection.

Explicit resolutions of double point singularities of surfaces

1999

Locally analytically, any isolated double point occurs as a double cover of a smooth surface. It can be desingularized explicitly via the canonical resolution, as it is very well-known. In this paper we explicitly compute the fundamental cycle of both the canonical and minimal resolution of a double point singularity and we classify those for which the fundamental cycle differs from the fiber cycle. Moreover we compute the conditions that a double point singularity imposes to pluricanonical systems. : 14J17, 32S25.

Deformations of algebroid stacks

Advances in Mathematics, 2011

In this paper we consider deformations of an algebroid stack on anétale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe.

VB-Courant Algebroids, E-Courant Algebroids and Generalized Geometry

Canadian Mathematical Bulletin, 2018

In this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids, and we construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra gl(V) ⊕ V correspond to complex Lie algebra structures on V.

Families and unfoldings of singular holomorphic Lie Algebroids

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE

In this paper, we investigate families of singular holomorphic Lie algebroids on complex analytic spaces. We introduce and study a special type of deformation called unfoldings of Lie algebroids, which generalizes the theory of singular holomorphic foliations developed by T. Suwa. We show that a one to one correspondence between transversal unfoldings and holomorphic flat connections on a natural Lie algebroid on the bases exists.

Embeddings of general blowups of abelian surfaces

manuscripta mathematica, 2000

Blowups of algebraic surfaces polarized by tensor powers of ample line bundles were studied by Coppens . In this note we complete the picture in the case of abelian surfaces studying blowups of surfaces polarized by primitive line bundles.

On the influence of the Segre Problem on the Mori cone of blown-up surfaces

2011

We propose a generalization of SHGH Conjectures to a smooth projective surface Y: the so called Segre Problem. The study of linear systems on Y can be translated in terms of the Mori cone of the blow up X = Blr Y at r general points. Generalizing a result from [dF10], we prove that if Segre Problem holds true, then a part of NE(X) does coincide with a part of the positive cone of X. NE(X) = Pos(X) + R(C), where Pos(X) is the positive cone and the sum runs on (−1)-curves. In order to generalize this kind of conjectures to any blown-up surface, we get interested in integral curves with negative self-intersection. We focus on a smooth projective surface Y and we transfer the study of linear systems of curves on Y passing through r general points x 1 ,. .. , x r with some multiplicities, to the study of curves on X = Bl r Y with negative self-intersection. We ask ourselves the natural generalized reformulation of SHGH conjectures: Problem. Let X = Bl r Y a blown-up surface at r general points; let us suppose h 2 (X , L) = 0 for all line bundles L associated to a non exceptional and non empty linear system L. If moreover L is reduced, then L is non special. This can easily seen to be false in a number of situations (see Section 3.2); the so called Segre Problem (Problem 3.7) is the refined statement of that problem. Since a consequence of the Segre Problem is the boundedness of negativity and arithmetic genus for the curves with negative self-intersection, we get to the statement of our main result: if the problem has a positive solution holds true, then a part of NE(X) is circular. Main Theorem. Let X = Bl r Y the blow up at r general points of a smooth projective surface Y and let L be the pullback to X of an ample A on Y. Let us suppose that for every integral curve C ⊂ X with negative self-intersection, C 2 −ν X and p a (C) π X. If r is large enough (explicit bounds depending only on A, ν X and π X), then there exists an explicit s ∈ R such that NE(X) (K −sL) 0 = Pos(X) (K −sL) 0 .