On the Saito basis and the Tjurina number for plane branches

2020 ◽  
Vol 373 (5) ◽  
pp. 3693-3707
Author(s):  
Y. Genzmer ◽  
M. E. Hernandes
Keyword(s):  
Tjurina Number

scholarly journals Freeness versus maximal global Tjurina number for plane curves

2016 ◽  
Vol 163 (1) ◽  
pp. 161-172 ◽  
Author(s):  
ALEXANDRU DIMCA
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
Tjurina Number ◽  
Free Plane

AbstractWe give a characterisation of nearly free plane curves in terms of their global Tjurina numbers, similar to the characterisation of free curves as curves with a maximal Tjurina number, given by A. A. du Plessis and C.T.C. Wall. It is also shown that an irreducible plane curve having a 1-dimensional symmetry is nearly free. A new numerical characterisation of free curves and a simple characterisation of nearly free curves in terms of their syzygies conclude this paper.

scholarly journals Splitting types of bundles of logarithmic vector fields along plane curves

2018 ◽  
Vol 29 (08) ◽  
pp. 1850055 ◽  
Author(s):  
Takuro Abe ◽  
Alexandru Dimca
Keyword(s):  
Vector Fields ◽  
Plane Curve ◽  
Plane Curves ◽  
Tjurina Number ◽  
Arrangements Of Lines ◽  
Splitting Type ◽  
Intersection Points

We give a formula relating the total Tjurina number and the generic splitting type of the bundle of logarithmic vector fields associated to a reduced plane curve. By using it, we give a characterization of nearly free curves in terms of splitting types. Several applications to free and nearly free arrangements of lines are also given, in particular a proof of a form of Terao’s Conjecture for arrangements having a line with at most four intersection points.

Milnor number and Tjurina number of complete intersections

1985 ◽  
Vol 271 (1) ◽  
pp. 121-124 ◽  
Author(s):  
Eduard Looijenga ◽  
Joseph Steenbrink
Keyword(s):  
Milnor Number ◽  
Complete Intersections ◽  
Tjurina Number

scholarly journals The minimal Tjurina number of irreducible germs of plane curve singularities

2021 ◽  
Vol 70 (4) ◽  
pp. 1211-1220
Author(s):  
Maria Alberich-Carraminana ◽  
Patricio Almiron ◽  
Guillem Blanco ◽  
Alejandro Melle-Hernandez
Keyword(s):  
Plane Curve ◽  
Tjurina Number ◽  
Plane Curve Singularities ◽  
Curve Singularities

scholarly journals Topological type of discriminants of some special families

2021 ◽  
Author(s):  
Evelia R. García Barroso ◽  
M. Fernando Hernández Iglesias
Keyword(s):  
Topological Type ◽  
Smooth Curve ◽  
Milnor Number ◽  
Irreducible Curve ◽  
Tjurina Number ◽  
Discriminant Curve ◽  
The Difference

AbstractWe will describe the topological type of the discriminant curve of the morphism $$(\ell , f)$$ ( ℓ , f ) , where $$\ell $$ ℓ is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.

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