Galindo, Carlos Intersections of 1-forms and valuations in a local regular surface. (English) Zbl 0832.14001 J. Pure Appl. Algebra 94, No. 3, 307-325 (1994). The author introduces intersection multiplicities between singular differential 1-forms in the plane and valuations centered at the plane. He proves a “Noether formula” showing the behaviour of the intersection multiplicity under quadratic transformations. Several methods to compute this number are derived. In the last part of the paper, intersection numbers are used to give a description of the desingularization of a 1-form without performing blow-ups. Reviewer: Carlos Galindo (Castellón) Cited in 1 Document MSC: 14B05 Singularities in algebraic geometry 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 32S65 Singularities of holomorphic vector fields and foliations Keywords:Noether formula; intersection multiplicities; singular differential 1-forms; quadratic transformations; desingularization PDFBibTeX XMLCite \textit{C. Galindo}, J. Pure Appl. Algebra 94, No. 3, 307--325 (1994; Zbl 0832.14001) Full Text: DOI References: [1] Abhyankar, S. S., On the valuations centered in a local domain, Amer. J. Math., 78, 321-348 (1956) · Zbl 0074.26301 [2] Cabrero, R., A criterion on formally jacobian rings (1993), University Valladolid: University Valladolid Spain, Preprint [3] Camacho, C.; Lins-Neto, A.; Sad, P., Topological invariants and equidesingularization for holomorphic vector fields, J. Differential. Geom., 20, 143-174 (1984) · Zbl 0576.32020 [4] Campillo, A., Algebroid curves in positive characteristics, (Lecture Notes in Mathematics, Vol. 613 (1980), Springer: Springer Berlin) · Zbl 0451.14010 [5] Cano, F., Desingularizations of plane vector fields, Trans. Amer. Math. Soc., 296, 1, 83-93 (1986) · Zbl 0612.14011 [6] Cano, F., Foliaciones singulares dicríticas, (Mem. Ac. Cienc. Exact. Fis. y Nat. Serie C. Exact. XXIV (1989)), Madrid, Spain [7] Galindo, C., Desarrollos de Hamburger-Noether y equivalencia discreta de valoraciones, (Thesis (1991), University of Valladolid: University of Valladolid Spain) [8] García, J., Valor de contacto y semigrupos de valores de una forma differencial (1990), University of Valladolid: University of Valladolid Spain, Preprint [9] Seidenberg, A., Reduction of singularities of the differential equation Adx + Bdy, Amer. J. Math., 90, 248-269 (1966) · Zbl 0159.33303 [10] Spivakovsky, M., Valuations in function fields of surfaces, Amer. J. Math., 112, 107-156 (1990) · Zbl 0716.13003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.