GEOMETRIC PROPERTIES OF AMPLE INVERTIBLE SHEAVES ON EXCEPTIONAL LOCI

This paper addresses the resolution of X\mathcal{X}X-dimensional Y\mathcal{Y}Y-singularity, focusing on the exceptional locus E\mathcal{E}E, where E\mathcal{E}E comprises irreducible components Ei\mathcal{E}_iEi isomorphic to Pn\mathbb{P}^nPn. These components are examined as invertible sheaves. The study investigates the conditions under which these sheaves are ample, utilizing Kleiman's Criterion (Kleiman, 1966) as a foundational tool. By applying this criterion, we determine the necessary and sufficient conditions for ampleness, offering insights into the geometric properties of the exceptional locus and contributing to the broader understanding of Y\mathcal{Y}Y-singularities and their resolutions