On infinitesimal deformations of singular varieties III

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.