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Is there a standard name for the type of singularities a codimension-$1$ subvariety of a smooth algebraic variety has when it looks locally (possibly analytically) like an arrangement of hyperplanes? «Normal crossings» refers to the special situation in which locally it looks like a subarrangement of the Boolean arrangement, that is, some coordinate hyperplanes.

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This paper uses the term "arrangement of smooth, complex algebraic hypersurfaces", or simply, "arrangement of smooth hypersurfaces". To quote: "Our goal here is to further generalize these results to a much wider class of arrangements of hypersurfaces, by which we mean a collection of smooth, irreducible, codimension 1 subvarieties which are embedded in a smooth, connected, complex projective algebraic variety, and which intersect locally like hyperplanes."

In another paper on the subject, Clément Dupont simply uses the term "hypersurface arrangement". To quote: "We develop a model for the cohomology of the complement of a hypersurface arrangement inside a smooth projective complex variety. This generalizes the case of normal crossing divisors, discovered by P. Deligne in the context of the mixed Hodge theory of smooth complex varieties."

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    $\begingroup$ the next sentence of the Abstract starts "If the hypersurfaces intersect locally like hyperplanes . . .", so presumably "arrangement of smooth hypersurfaces" does not imply this condition. $\endgroup$ – Noam D. Elkies Dec 1 '17 at 23:17
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    $\begingroup$ This may not be crystal clear from the abstract, but it gets specified in the intro. I added a fuller quote, hopefully this will clarify the meaning. $\endgroup$ – Alex Suciu Dec 2 '17 at 2:35

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