On infinitesimal deformations of singular varieties III

TL;DR

This paper investigates the infinitesimal deformations of a specific singular variety, the affine cone over a reducible nodal curve formed by gluing three projective lines, analyzing its embedding and singular locus.

Contribution

It provides a detailed analysis of the affine cone over a reducible nodal curve, including its embedding via a very ample line bundle and the structure of its singular locus.

Findings

The line bundle of multidegree (4,3,3) is very ample, enabling an embedding into projective space.

The affine cone's singular locus consists of three lines meeting at the vertex.

The structure of the singularities is explicitly characterized.

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.