Contact loci and Hironaka's order

Abstract

We study contact loci sets of arcs and the behavior of Hironaka's order function defined in constructive Resolution of singularities. We show that this function can be read in terms of the irreducible components of the contact loci sets at a singular point of an algebraic variety. ContentsIntroduction 1 1. Arcs, valuations and contac loci 5 2. Nash multiplicity sequences, the persistance, and the Q-persistance 8 3. Rees algebras 11 4. Local presentations of the Multiplicity 13 5. Hironaka's order function, the persistance, and the Q-persistance 15 6. Generic values in contact loci sets 17 7. Fat irreducible components of contact loci and Hironaka's order 20 8. On the values of the Q-persistance 24 References 25

Finite morphisms and Nash multiplicity sequences

“…A consequence of Theorem 1.1 is that ord (d) X (ξ) can be defined without using étale topology and only studying properties of its space of arcs. Moreover the arc η realizing the minimum in (1.1.2) can be choosen, and constructed explicitly, being fat and divisorial [9,Theorem 6.3]. In other words, the refinement of the multiplicity for the resolution function in (1.0.4) can be obtained by studying sequences of Nash multipicities sequences in L(X).…”