A note on geometric degree of finite extensions of mappings from a smooth variety

• Autorzy: Karaś M.
• Języki publikacji: EN

Abstrakty

EN

Let ƒ : V → W be a finite polynomial mapping of algebraic subsets V, W of k[sup]n and k[sup]m, respectively, with n ≤ m. Kwieciński [J. Pure Appl. Algebra 76 (1991)] proved that there exists a finite polynomial mapping F : k[sup]n → k[sup]m such that F| v = ƒ. In this note we prove that, if V, W ⊂ k[sup] are smooth of dimension k with 3k + 2 ≤ n, and ƒ : V → W is finite, dominated and dominated on every component, then there exists a finite polynomial mapping F : k[sup]n → [sup]n such that F|v = ƒ and gdeg F ≤ (gdegƒ)[sup]k+1. This improves earlier results of the author.

Słowa kluczowe

EN

finite mapping; geometric degree; extension of mappings

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2008
• Tom: Vol. 56, no 2
• Strony: 105--108
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Karaś M.

• Instytut Matematyki, Uniwersytet Jagielloński, Reymonta 4, 30-059 Kraków, Poland,

Bibliografia

• [1] Z. Jelonek, The extension of regular and rational embeddings, Math. Ann. 277 (1987), 113-120.
• [2] -, A note about the extension of polynomial embeddings, Bull. Polish Acad. Sci. Math. 43 (1995), 239-244.
• [3] S. Kaliman, Extension of isomorphisms between affine algebraic subvarieties of kn to automorphisms of kn, Proc. Amer. Math. Soc. 113 (1991), 325-334.
• [4] M. Karaś, An estimation of the geometric degree of an extension of some polynomial proper mappings, Univ. Iagell. Acta Math. 35 (1997), 131-135.
• [5] M. Karaś, Geometric degree of finite extension of projections, ibid. 37 (1999), 109-119.
• [6] -, Birational finite extensions, J. Pure Appl. Algebra 148 (2000), 251-253.
• [7] -, Finite extensions of mappings from a smooth variety, Ann. Polon. Math. 75 (2000), 79-86.
• [8] -, Finite extensions of mappings of finite sets, Bull. Polish Acad. Sci. Math. 50 (2002), 239-241.
• [9] -, Geometric degree of finite extension of mappings from a smooth variety, J. Pure Appl. Algebra 212 (2008), 1145-1148.
• [10] M. Kwieciński, Extending finite mappings to affine space, ibid. 76 (1991), 151-153.
• [11] V. Srinivas, On the embedding dimension of the affine variety, Math. Ann. 289 (1991), 125-132.