Unfolding plane curves with cusps and nodes release_hykut23drrgyrcrtlbsqj6efuu

Abstract

Given an irreducible surface germ (<jats:italic>X</jats:italic>, 0) ⊂ (ℂ<jats:sup>3</jats:sup>, 0) with a one-dimensional singular set<jats:italic>Σ</jats:italic>, we denote by<jats:italic>δ</jats:italic><jats:sub>1</jats:sub>(<jats:italic>X</jats:italic>, 0) the delta invariant of a transverse slice. We show that<jats:italic>δ</jats:italic><jats:sub>1</jats:sub>(<jats:italic>X</jats:italic>, 0) ≥<jats:italic>m</jats:italic><jats:sub>0</jats:sub>(<jats:italic>Σ</jats:italic>, 0), with equality if and only if (<jats:italic>X</jats:italic>, 0) admits a corank 1 parametrization<jats:italic>f</jats:italic>:(ℂ<jats:sup>2</jats:sup>, 0) → (ℂ<jats:sup>3</jats:sup>, 0) whose only singularities outside the origin are transverse double points and semi-cubic cuspidal edges. We then use the local Euler obstruction Eu(<jats:italic>X</jats:italic>, 0) in order to characterize those surfaces that have finite codimension with respect to<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S0308210513000632_inline1" xlink:type="simple" />-equivalence or as a frontal-type singularity.
In application/xml+jats format

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