Question

Algebraic Geometry Robin Hartshorne: Blowing Up Curve Singularities Part (d):

5.6. Blowing Up Cure Singularities. (a) Let Y be the cusp or node of (Ex. 5.1). Show that the curve 1. obtained b;y blowing up Y at O = (0.0) is nonsingular (cf. (4.9.1) and (Ex. 4.10) . (b) We define a node (also called ordinary douhle poinı) to be a double point (i.e., a point of multiplicity 2) of a plane curve with distinct tangent directions (Ex. 5.3). If P is a node on a plane curve Y, show that φ-1(P) consists of two distinct nonsingular points on the blown-up curve Y. We say that "blowing up P resolves the singularity at P" (c) Let Pe Y be the tacnode of (Ex. 5.1) IfpY is the blowing-up at P. show that p P) is a node. Using (b) we see that the tacnode can be resolved by two successive blowings-up (d) Let Y be the plane curvehich has a "higher order cusp" at O. Show that O is a triple point; that blowing up O gives rise to a double point (what kind?) and that one further blowing up resolves the singularity Note: We will see later (V. 3.8) that any singular point of a plane curve can be resolved by a finite sequence of successive blowings-up

Answer 1

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