2015/07/31 by Orlando E. Villamayor U, U, Orlando E. Villamayor
Computer Science · Mathematics · Medicine · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Magnolia and Illicium research #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1507.08948
openalex publication_date 2015/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Resolution of singularities of varieties over fields of characteristic zero\ncan be proved by using the multiplicity as main invariant. The proof of this\nresult leads to new questions in positive characteristic. We discuss here\nresults which follow by induction on the dimension of the varieties. Fix a\nvariety X(d) of dimension d over a em perfect field k or, more\ngenerally, a pure dimensional scheme of finite type over k. Fix a closed\npoint x\∈ X(d) of multiplicity e>1. Define a local simplification of\nthe multiplicity at x\∈ X(d) as a proper birational map, say\nX(d)\← X(d)1, where X(d) denotes now a neighborhood of\nx, so that X(d)1 has multiplicity <e at any point x1\∈ X(d)1.\n Assume, by induction on d, the existence of local simplifications of the\nmultiplicity for schemes over k of dimension d', for all d' <d. We prove,\nunder this inductive assumption, that a local simplification at x\∈ X(d)\ncan be constructed when (CX,x)red is not regular.\n Here CX,x denotes the tangent cone of x\∈ X, and (CX,x)red is\nthe reduced scheme. The paper uses classical results of commutative algebra,\nand compares the effect of blowing up along equimultiple centers, and along\nnormally flat centers.\n