Theory and Reduction of Singularities

Let x₁, x₂,…, x_r+1 be homogeneous point coördinates in a complex projective γ-dimensional space S_r. An algebraic variety V in S _γ is the locus of point (x) satisfiying a system of algebraic equations,1$${f_1}\left( {{x_{1,}} \ldots ,{x_\gamma }_{ + 1}} \right) = 0, \ldots ,{f_n}\left( {{x_1}, \ldots ,{x_{\gamma + 1}}} \right) = 0$$, where f₁, f₂,…, f_n are homogeneous polynomials. If ϕ is a homogenous polynomial in the x’s which vanishes at all the common zeros of f₁, …, f_n, i. e. at every point of V, we say briefly that ϕ vanishes (ϕ= 0) on V. The variety V is irreducible, if from ϕ ψ = 0 on V it follows necessarily that one at least of the polynomials ϕ, ψ vanishes on V. In the language of the theory of ideals this definition can be formulated as follows: V is irreducible if the homogeneous polynomial ideal (f₁,…f_n) (H-ideal) is a primary ideal1 (Macaulay, a, p. 33; van der Waerden, a₂, p. 54). From the theorem of Hilbert-Netto (Macaulay, a, p. 48; van der Waerden, a₂, p. 11 ) it follows then that either ϕϱ or ψϱ is a member of (f₁,…, f_n), where ϱ is a convenient integer.