Let Z be a d-dimensional subscheme which can be embedded into a smooth variety A of dimension n. Let S(Z) be the singular points of Z, its complement in Z. Let be the Grassmannian bundle of d-dimensional subspaces of the tangent bundle TA over A.
Consider the morphism defined by
where is the tangent space of Z at z. Denote the closure of in . Then the morphism
induced by the first projection (locally ) is called the Nash blowing-up of . Note that the construction of is independent of the embedding up to unique -isomorphism.
Let such that . Now let’s describe the fan of the weighted projective space . Define the lattice where , are standard basis vectors in . Then the fan is made up of the cones generated by all the proper subsets of Continue reading →