The relation between toric geometry, theory of local blow-ups and ramification theory and their applications in valuation theory

• Grant number: 17/17835-9
• Support Opportunities: Research Grants - Young Investigators Grants
• Start date: March 01, 2018
• End date: February 29, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Josnei Antonio Novacoski
• Grantee: Josnei Antonio Novacoski
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• City of the host institution: São Carlos
• Associated research grant(s): 25/10540-0 - On current problems in positive characteristic, AP.JP2; 22/14876-4 - The defect of an extension of valued fields, AV.EXT
• Associated scholarship(s): 20/05148-0 - The valuative tree, BP.MS; 18/23727-7 - Topologies on spaces of valuations, BP.MS; 18/04068-2 - The relation between toric geometry, theory of local blow-ups and ramification theory and their applications to valuation theory, BP.JP

Abstract

The main goal of this research project is to expand the knowledge of the relation between toric geometry, theory of local blow-ups and ramification theory. The motivation for this study is the common applications of these areas to valuation theory, more specifically to the local uniformization problem. The local uniformization problem (that can be seen as a local version of resolution of singularities) is open for valuations centered on algebraic varieties over a field of positive characteristic. However, in the recent years, many programs to solve this problem have gainned strength. Three of the most important are those developed by Teissier (using toric geometry), by Spivakovsky (using theory of local blow-ups) and by Knaf and Kuhlmann (using ramification theory). The problems which will be studied in this project will allow us to adapt strategies from each of these programs to the others. This will provide new results in the problem of local uniformization in positive characteristic. (AU)

Scientific publications

(15)

(The scientific publications listed on this page originate from the Web of Science or SciELO databases. Their authors have cited FAPESP grant or fellowship project numbers awarded to Principal Investigators or Fellowship Recipients, whether or not they are among the authors. This information is collected automatically and retrieved directly from those bibliometric databases.)

BARNABE, MATHEUS DOS SANTOS; NOVACOSKI, JOSNEI. Valuations on K[x] approaching a fixed irreducible polynomial. Journal of Algebra, v. 592, p. 100-117, FEB 15 2022. (17/17835-9)

BARNABE, M. S.; NOVACOSKI, J.; SPIVAKOVSKY, M.. On the structure of the graded algebra associated to a valuation. Journal of Algebra, v. 560, p. 667-679, OCT 15 2020. (17/17835-9)

CUTKOSKY, STEVEN DALE; NOVACOSKI, JOSNEI. Essentially finite generation of valuation rings in terms of classical invariants. Mathematische Nachrichten, v. 294, n. 1, p. 15-37, JAN 2021. (17/17835-9)

NOVACOSKI, JOSNEI; SPIVAKOVSKY, MARK. Kähler differentials, pure extensions and minimal key polynomials. Israel Journal of Mathematics, v. N/A, p. 41-pg., 2025-11-18. (17/17835-9)

DE MORAES, MICHAEL; NOVACOSKI, JOSNEI. Limit key polynomials as p-polynomials. Journal of Algebra, v. 579, p. 22-pg., 2021-04-02. (17/17835-9)

NOVACOSKI, JOSNEI. Key polynomials and minimal pairs. Journal of Algebra, v. 523, p. 1-14, APR 1 2019. (17/17835-9, 15/23409-7)

NOVACOSKI, J. A.; SILVA DE SOUZA, C. H.. On truncations of valuations. Journal of Pure and Applied Algebra, v. 226, n. 6, JUN 2022. (17/17835-9, 20/05148-0)

KUHLMANN, FRANZ-VIKTOR. Valued fields with finitely many defect extensions of prime degree. JOURNAL OF ALGEBRA AND ITS APPLICATIONS, v. 21, n. 03, p. 18-pg., 2022-03-01. (17/17835-9)

NOVACOSKI, JOSNEI. On MacLane-Vaquie key polynomials. Journal of Pure and Applied Algebra, v. 225, n. 8, AUG 2021. (17/17835-9)

DE MORAES, MICHAEL; NOVACOSKI, JOSNEI. Perron transforms and Hironaka's game. Journal of Algebra, v. 563, p. 100-110, SEP 1 2020. (17/17835-9)

NART, ENRIC; NOVACOSKI, JOSNEI. Geometric parametrization of valuations on a polynomial ring in one variable. MATHEMATISCHE ZEITSCHRIFT, v. 304, n. 4, p. 21-pg., 2023-08-01. (21/11246-7, 17/17835-9)

NART, ENRIC; NOVACOSKI, JOSNEI. Minimal limit key polynomials. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, v. 111, n. 5, p. 34-pg., 2025-05-01. (17/17835-9)

NOVACOSKI, JOSNEI; SILVA DE SOUZA, CAIO HENRIQUE. Parametrizations of subsets of the space of valuations. MATHEMATISCHE ZEITSCHRIFT, v. 307, n. 4, p. 25-pg., 2024-08-01. (21/13531-0, 17/17835-9)

NART, ENRIC; NOVACOSKI, JOSNEI. The defect formula. ADVANCES IN MATHEMATICS, v. 428, p. 44-pg., 2023-06-12. (21/11246-7, 17/17835-9)

NOVACOSKI, JOSNEI. Generators for extensions of valuation rings. Journal of Pure and Applied Algebra, v. 229, n. 2, p. 13-pg., 2025-02-01. (17/17835-9)