Monotonic invariants under blowups

Abstract

We prove that the numerical invariant [Formula: see text] of a reduced irreducible plane curve singularity germ is non-negative, non-decreasing under blowups and strictly increasing unless the curve is non-singular. This provides a new perspective to understand the question posed by Dimca and Greuel. Moreover, our work can be put in the general framework of discovering monotonic invariants under blowups.

On the Approximate Polar Curves of Foliations

“…In [6, Question 4.2], Dimca and Greuel present a family of curves {C m } m≥1 such that the quotient µ (Cm) τ (Cm) is strictly increasing with limit 4/3 and they ask if this property is true for any plane curve singularity. The question was affirmatively answered in [1], [10], and [15] for the irreducible case and in [2] for the reduced case.…”

On the quotient of Milnor and Tjurina numbers for two‐dimensional isolated hypersurface singularities

“…Despite the fact that both papers use quite different techniques, both are based on the explicit computations about the moduli space of an irreducible plane curve singularity given by Genzmer in [10]. Finally, Wang in [41] gives another alternative proof for the irreducible case based also in Genzmer's result about the dimension of the generic component of the moduli space [10]. Moreover, Wang's approach is of different nature since he proves that $$3\mu -4\tau$$ satisfy a certain property (monotonicity under blow ups) which provides a nice perspective in the possible applications of Dimca and Greuel's Question 1.1 in the irreducible case.…”

“…Thus, we provide a solution to Problem 1 in the case 𝑁 = 2, 𝑟 = 1. Moreover, as one can see, our point of view is completely new from the techniques used in [1,2,11,41] to solve Question 1.1 for some special cases.…”

“…There are some partial answers to Dimca and Greuel's Question 1.1 given by Blanco and the author [2], Alberich‐Carramiñana, Blanco, Melle‐Hernández and the author [1], Genzmer and Hernandes [11] and Wang [41] (see Section 3 for a more detailed description of those results). All of them show that $$\mu /\tau$$ satisfies the 4/3 bound for some special families of plane curve singularities.…”

“…All of them show that $$\mu /\tau$$ satisfies the 4/3 bound for some special families of plane curve singularities. Despite those partial positive answers to Question 1.1, there is no clue in Wang's results [41] nor in Genzmer–Hernandes [11] neither in our first results [1, 2] as to whether the numbers 3, 4 can be inferred from deeper arguments than just explicit computations, i.e., is there an intrinsic reason to consider the invariant $$3\mu -4\tau$$ instead of any other combination of the type $$a\mu -b\tau$$ with $$(a,b)\ne (3,4)?$$…”