Finite extensions of mappings from a smooth variety

ArticleOriginal scientific text

Title

Authors ¹

Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

Let V, W be algebraic subsets of

k^{n}

,

k^{m}

respectively, with n ≤ m. It is known that any finite polynomial mapping f: V → W can be extended to a finite polynomial mapping

F : k^{n} \to k^{m} .

The main goal of this paper is to estimate from above the geometric degree of a finite extension

F : k^{n} \to k^{n}

of a dominating mapping f: V → W, where V and W are smooth algebraic sets.

finite extension, geometric degree, finite mapping

S. S. Abhyankar, On the Semigroup of a Meromorphic Curve, Kinokuniya Book-Store, Tokyo, 1978.
Z. Jelonek, The extension of regular and rational embeddings, Math. Ann. 277 (1987), 113-120.
M. Karaś, An estimation of the geometric degree of an extension of some polynomial proper mappings, Univ. Iagell. Acta Math. 35 (1997), 131-135.
M. Karaś, Geometric degree of finite extensions of projections, ibid. 37 (1999), 109-119.
M. Karaś, Birational finite extensions, J. Pure Appl. Algebra 148 (2000), 251-253.
M. Kwieciński, Extending finite mappings to affine space, ibid. 76 (1991), 151-153.
D. Mumford, Algebraic Geometry I. Complex Projective Varieties, Springer, Heidelberg, 1976.
K. J. Nowak, The extension of holomorphic functions of polynomial growth on algebraic sets in $ℂ^{n}$ , Univ. Iagell. Acta Math. 28 (1991), 19-28.
I. R. Shafarevich, Basic Algebraic Geometry, Springer, Heidelberg, 1974.