Finite morphisms and simultaneous reduction of the multiplicity

Abstract

Let X be a singular algebraic variety defined over a field k, with quotient field K(X). Let s ≥ 2 be the highest multiplicity of X and Fs(X) the set of points of multiplicity s. If Y ⊂ Fs(X) is a regular center and X ← X1 is the blow up at Y , then the highest multiplicity of X1 is less than or equal to s. A sequence of blow ups at regular centers Yi ⊂ Fs(Xi), say X ← X1 ← · · · ← Xn, is said to be a simplification of the multiplicity if the maximum multiplicity of Xn is strictly lower than that of X, that is, if Fs(Xn) is empty. In characteristic zero there is an algorithm which assigns to each X a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive.In this paper we will study finite dominant morphisms between singular varieties β : X ′ → X of generic rank r ≥ 1 (i.e., [K(X ′ ) : K(X)] = r). We will see that, when imposing suitable conditions on β, there is a strong link between the strata of maximum multiplicity of X and X ′ , say Fs(X) and Frs(X ′ ) respectively. In such case, we will say that the morphism is strongly transversal. When β : X ′ → X is strongly transversal one can obtain information about the simplification of the multiplicity of X from that of X ′ and vice versa. Finally, we will see that given a singular variety X and a finite field extension L of K(X) of rank r ≥ 1, one can construct (at least locally, inétale topology) a strongly transversal morphism β : X ′ → X, where X ′ has quotient field L.2010 Mathematics Subject Classification. 14E15.

Nash multiplicity sequences and Hironaka's order function

“…When the characteristic is positive, G (d) i is generated by suitable symmetric polynomial functions evaluated on the coefficients of the f i (cf. [46], [47,§1,Definition 4.10]). Now we claim that β : V (n) → V (d) is G (n) -admissible and that, up to integral closure,…”

“…When the characteristic is zero, resolving G (2) is equivalent to resolving G (1) and the fact that one single blow up is enough is reflected in the value 3/2: G (1) is resolved in one step.…”

“…When the characteristic is 2, H (1) somehow exagerates the image of the singular locus of H (2) : it takes two blow ups to resolve H (1) while H (2) is resolved by one. And yet, there is no other S-Rees algebra that approximates the image of the singular locus of H (2) than H (1) . Thus: why the value ord…”

“…Thus, in the characteristic 2 case, it takes longer to separate the graph of the arc from the maximum multiplicity locus of X × A 1 k and the order of H (1) at the origin is reflecting this fact: this order cannot take a value below 2.…”

“…In both cases the natural inclusion S = k[x] ⊂ R = k[x, y] is admissible. The elimination algebra for G (2) is G (1) = S[x 2 W, x 3 W 2 ], and the one for H(2) is H(1) = S[x 2 W ]. Thus we have two different values for Hironaka's order function in dimension 1 = dim k (X), depending on the characteristic: ord…”