On certain type of singular varieties with smooth subvarieties

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).