On the minimal value of global Tjurina numbers for line arrangements
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On the minimal value of global Tjurina numbers for line arrangements Alexandru Dimca1,2 Dedicated to the memory of my dear friend S¸ tefan Papadima Received: 16 January 2019 / Revised: 11 July 2019 / Accepted: 11 September 2019 © Springer Nature Switzerland AG 2019
Abstract We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by Andrew A. du Plessis and Charles T. C. Wall, can be improved when the curve is a line arrangement. This fact is in sharp contrast to a conjecture saying that the general upper bound for the global Tjurina number of a reduced complex projective plane curve, also given by du Plessis and Wall, is realized by line arrangements in practically all cases. Keywords Tjurina number · Milnor number · Line arrangement Mathematics Subject Classification 14H50 · 14B05 · 32S22
1 Introduction and main results Let S = C[x, y, z] be the graded polynomial ring in three variables x, y, z and let C : f = 0 be a reduced curve of degree d in the complex projective plane P2. The minimal degree of a Jacobian relation for f is the integer mdr( f ) defined to be the smallest integer m 0 such that there is a nontrivial relation a fx + b f y + c fz = 0
This work was partially supported by the French government, through the UCAJEDI Investments in the Future Project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01 and by the Romanian Ministry of Research and Innovation, CNCS - UEFISCDI, Grant PN-III-P4-ID-PCE-2016-0030, within PNCDI III.
B
Alexandru Dimca [email protected]
1
Université Côte d’Azur, CNRS, LJAD, Nice, France
2
Simion Stoilow Institute of Mathematics, P.O. Box 1-764, 014700 Bucharest, Romania
123
A. Dimca
among the partial derivatives f x , f y and f z of f with coefficients a, b, c in Sm , the vector space of homogeneous polynomials of degree m. We say that the plane curve has type (d, r ) if d = deg( f ) and r = mdr( f ). When mdr( f ) = 0, then C is a pencil of lines, i.e., a union of lines passing through one point, a situation easy to analyze. We assume from now on that mdr( f ) 1. Denote by τ (C) the global Tjurina number of the curve C, which is the sum of the Tjurina numbers of the singular points of C. When C is a line arrangement, its global Tjurina number coincides with its global Milnor number μ(C), and is given by τ (C) =
(n( p) − 1)2,
(1.1)
p
the sum being over all multiple points p of C, and n( p) denoting the multiplicity of C at p. In this note we consider the minimal values of τ (C) when C : f = 0 is a line arrangement, once we fix its type (d, r ). Let m(C) be the maximal multiplicity of a point in C, and n(C) the maximal multiplicity of a point in C \ { p}, where p is any point in C of multiplicity m(C). Note that 1 n(C) m(C) d. Moreover m(C) = d if and only if mdr( f ) = 0, and m(C) = d − 1 if and only if mdr( f ) = 1, see [8, Proposition 4.7]. In addition, the case 2 = n(C) m(C) d −2 corresponds to the intersection lattice L(C) being the lattice L(d, m(C))
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