Finite morphisms and simultaneous reduction of the multiplicity

Open Access

Abstract

"This is the peer reviewed version of the following article: Abad C, Bravo A, Villamayor U. OV. Finite morphisms and simultaneous reduction of the multiplicity. Mathematische Nachrichten. 2020;293:8–38, which has been published in final form at https://doi.org/10.1002/mana.201800470. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley's version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited" ; Let X be a singular algebraic variety defined over a perfect field k, with quotient field 𝐾�(𝑋�). Let 𝑠�≥2 be the highest multiplicity of X and let 𝐹�𝑠�(𝑋�) be the set of points of multiplicity s. If 𝑌�⊂𝐹�𝑠�(𝑋�) is a regular center and 𝑋�←𝑋�1 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 𝑌�𝑖�⊂𝐹�𝑠�(𝑋�𝑖�), say 𝑋�←𝑋�1←⋯←𝑋�𝑛�, is said to be a simplification of the multiplicity if the maximum multiplicity of 𝑋�𝑛� is strictly lower than that of X, that is, if 𝐹�𝑠�(𝑋�𝑛�) 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 𝐹�𝑠�(𝑋�) and 𝐹�𝑟�𝑠�(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, in étale topology) a strongly transversal morphism 𝛽�:𝑋�′→𝑋�, where 𝑋�′ has quotient field L. ; The authors were partially supported by the Spanish Ministry of Economy and Competitiveness, through the "Severo Ochoa" Programme for Centres of Excellence in R&D (SEV2015-0554), and through MTM2015-68524-P (MINECO/FEDER)