On infinitesimal deformations of singular varieties III

Abstract

We study the affine cone over a reducible nodal curve $X$ obtained by gluing three projective lines along three pairs of points to form a connected curve of arithmetic genus \(1\). We endow \(X\) with a line bundle \(L\) of multidegree \((4,3,3)\), and we show that \(L\) is very ample, giving an embedding into \( \mathbb{P}^9\). We then analyze in detail the affine cone \( C(X) \) and determine its singular locus, which consists of three singular lines meeting at the vertex.