2014/12/16 by Hong Duc Nguyen, Nguyen, Hong Duc
Arts and Humanities · Mathematics · Social Sciences · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Communism, Protests, Social Movements #FOS: Mathematics #French Historical and Cultural Studies
paper · pdf · doi:10.48550/arxiv.1412.5007
openalex publication_date 2014/12/16 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We study classical invariants for plane curve singularities f\∈ K[[x,y]],\nK an algebraically closed field of characteristic p\≥ 0: Milnor number,\ndelta invariant, kappa invariant and multiplicity. It is known, in\ncharacteristic zero, that \μ(f)=2\δ(f)-r(f)+1 and that\n\κ(f)=2\δ(f)-r(f)+\mt(f). For arbitrary characteristic,\nDeligne prove that there is always the inequality \μ(f)\≥\n2\δ(f)-r(f)+1 by showing that \μ(f)-\( 2\δ(f)-r(f)+1\)\nmeasures the wild vanishing cycles. By introducing new invariants\n\γ,\\γ, we prove in this note that \κ(f)\≥\n\γ(f)+\mt(f)-1\≥ 2\δ(f)-r(f)+\mt(f) with equalities\nif and only if the characteristic p does not divide the multiplicity of any\nbranch of f. As an application we show that if p is "big" for f (in fact\np > \κ(f)), then f has no wild vanishing cycle. Moreover we obtain some\nPl "ucker formulas for projective plane curves in positive characteristic.\n