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On certain type of singular varieties with smooth subvarieties

Research paper by M. R. Gonzalez-Dorrego

Indexed on: 02 Jan '20Published on: 02 Jan '20Published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas



Abstract

Let k an algebraically closed field, \(char\ k=0.\) Let C be an irreducible nonsingular curve such that \(kC=S\cap X\), \(k\in {\mathbb {N}},\) where S and X are two surfaces and all the singularities of X are of the form \(z^{p}=x^{s}-y^{s},\)\(p, \, s\) primes, \(s=pt+m,\)\(t\in {\mathbb {N}}\), \(m=1\) or 2. We also study the cases, for \(p, \, s\) primes, \(p=4r+1,\)\(r \in {\mathbb {N}},\)\(s=pt+2r+1,\)\(t \in {\mathbb {N}},\) and \(p=4r+3,\)\(r \in {\mathbb {N}},\)\(s=pt+2r+2,\)\(t \in {\mathbb {N}}.\) We prove that C can never pass through such kind of singularities of a surface, unless \(k=pa,\)\(a\in {\mathbb {N}}\). We study multiplicity-k structures on varieties, \(k\in {\mathbb {N}}.\) Let Z be a reduced irreducible nonsingular \((n-1)\)-dimensional variety such that \(kZ=X\cap S\), where S is a \((N-1)\)-fold in \({\mathbb {P}}^{N},\)X is a normal n-fold with certain type of singularities, like linear compound \(V_{ps}\) singularity or (d,l) complete intersection compound \(V_{ps}\) singularity, We study when \(Z\cap \text {Sing} (X)\ne \emptyset .\) These results generalize some results in Gonzalez-Dorrego (On singular varieties with smooth subvarieties, singularities in geometry, topology, foliations and dynamics. Birkhauser, Basel, pp 125–134, 2017). Seifert invariants of \(z^{p}=x^{s}-y^{s},\)\(p, \, s\) primes, \(s=pt+m,\)\(t\in {\mathbb {N}}\), \(m=1,\) 2 and for \(p=4r+1,\)\(r \in {\mathbb {N}},\)\(s=pt+2r+1,\)\(t \in {\mathbb {N}},\) and\(p=4r+3,\)\(r \in {\mathbb {N}},\)\(s=pt+2r+2,\)\(t \in {\mathbb {N}}.\) are studied (Prop. 34).