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Analytic and Algebraic Geometry - ebook/pdf
Analytic and Algebraic Geometry - ebook/pdf
Autor: , Liczba stron: 202
Wydawca: Wydawnictwo Uniwersytetu Łódzkiego Język publikacji: polski
ISBN: 978-8-3796-9243-9 Data wydania:
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Kategoria: ebooki >> naukowe i akademickie
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 Annual Conferences in Analytic and Algebraic Geometry have been organized by Faculty of Mathematics and Computer Science of the University of Łódź since 1980. Until now, proceedings of these conferences (mainly in Polish) have comprised educational materials describing current state of a branch of mathematics, new approaches to known topics, and new proofs of known results (http://konfrogi.math.uni.lodz.pl/).
The volume include new results and survey articles concerning real and complex alge-braic geometry, singularities of curves and hypersurfaces, invariants of singularities (the Milnor number, degree of C0-sufficiency), algebraic theory of derivations and others topics.
English translation of the Polish version of an article by Stanisław Łojasiewicz (1926-2002) devoted to the famous Hironaka theorem on resolution of singularities. It contains his original approach to the problem in the case of curves and coherent analytic sheaves on 2-dimensional manifolds. This interesting article has not yet been available in English.
The volume is dedicated to the memory of Stanisław Łojasiewicz. We are deeply indebt-ed to him for introducing the topics of analytic and algebraic geometry in the Łódź mathematical centre.
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Tadeusz Krasiński – Department of Algebraic Geometry and Theoretical Computer Science, Faculty of Mathematics and Computer Science, 90-238 Łódź, Banacha Str. 22 krasinski@uni.lodz.pl Stanisław Spodzieja – Department of Analytical Functions and Differential Equations, Faculty of Mathematics and Computer Science, 90-238 Łódź, Banacha Str. 22 spodziej@math.uni.lodz.pl COVER DESIGN Michał M. Jankowski © Copyright by University of Łódź, Łódź 2013 Publication reviewed Printed directly from camera‐ready materials provided to the Łódź University Press by Department of Analytical Functions and Differential Equations First Edition. W.06395.13.0.K ISBN (wersja drukowana) 978-83-7969-017-6 ISBN (ebook) 978-83-7969-243-9 Łódź University Press 90-131 Łódź, ul. Lindleya 8 www.wydawnictwo.uni.lodz.pl e-mail: ksiegarnia@uni.lodz.pl tel. (42) 665 58 63, faks (42) 665 58 62 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Photography of Stanisław Łojasiewicz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Facsimile of one page by Stanisław Łojasiewicz. . . . . . . . . . . . . . . . . . . . . . .9 1. Stanisław Łojasiewicz Geometric desingularization of curves in manifolds . . . . . . . . . . . . . . . . . . 11 2. Szymon Brzostowski, Necessary conditions for irreducibility of algebroid plane curves . . . . 33 3. Evelia Rosa Garc´ıa Barroso and Arkadiusz Płoski, Euclidean algorithm and polynomial equations after Labatie . . . . . . . 41 4. Zbigniew Jelonek, On smooth hypersurfaces containing a given subvariety . . . . . . . . . . . . 51 5. Piotr Jędrzejewicz, Rings of constants of polynomial derivations and p-bases . . . . . . . . . . . 57 6. Grzegorz Oleksik, On combinatorial criteria for isolated singularities . . . . . . . . . . . . . . . . . . 81 7. Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław Spodzieja, On C0-sufficiency of jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95 8. Arkadiusz Płoski, Introduction to the local theory of plane algebraic curves . . . . . . . . . . 115 9. Jean-Marie Strelcyn, On Chouikha’s isochronicity criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10. Justyna Walewska, The jump of Milnor numbers in families of non-degenerate and non-convenient singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 11. Michał Zakrzewski, Henryk Żołądek, Multiple zeta values and the WKB method . . . . . . . . . . . . . . . . . . . . . . . 155 Preface Annual Conferences in Analytic and Algebraic Geometry have been organized by Faculty of Mathematics and Computer Science of the University of Łódź since 1980. Until now, proceedings of these conferences (mainly in Polish) have comprised educational materials describing current state of a branch of mathematics, new approaches to known topics, and new proofs of known results (see the Internet page: http://konfrogi.math.uni.lodz.pl/). The subject of the present volume include new results and survey articles concer- ning real and complex algebraic geometry, singularities of curves and hypersurfaces, invariants of singularities (the Milnor number, degree of C0-sufficiency), algebraic theory of derivations and others topics. One remarkable element of this collection is an English translation of the Polish version, published in proceedings of the above mentioned conferences, of an article by Stanisław Łojasiewicz (1926-2002) devoted to the famous Hironaka theorem on resolution of singularities. It contains his original approach to the problem in the case of curves and coherent analytic sheaves on 2-dimensional manifolds. This interesting article has not yet been available in English. Additionally, we add a photo portrait of him and the facsimile of one page of his original handwritten manuscript. We would like to thank Arkadiusz Płoski for the help in preparing the volume, Michał Jankowski for designing the cover, referees for preparing reports of the articles and all participants of the Conferences for their good humor and enthusiasm in doing mathematics. Finally, we would like to thank Stanisław Łojasiewicz jr and Anna Ostoja- Łojasiewicz, the heirs of Stanisław Łojasiewicz, for having agreed to include his article into this volume. We dedicate the whole volume to the memory of Stanisław Łojasiewicz. November 2013, Łódź Tadeusz Krasiński Stanisław Spodzieja Stanisław Łojasiewicz (9 X 1926 – 14 XI 2002) (The photo was taken by Przemysław Skibiński in 2000) The facsimile of the first page of the Polish handwritten version of the article (1988) by Stanisław Łojasiewicz, translated in this volume. Analytic and Algebraic Geometry Łódź University Press 2013, 11 – 32 GEOMETRIC DESINGULARIZATION OF CURVES IN MANIFOLDS ∗) ∗∗) STANISŁAW ŁOJASIEWICZ 1. Introduction The article does not pretend to any originality. In the literature there exists a number of descriptions of desingularizations in the case of curves. Deciding for this description the author think it is worth looking in details into this fascinating topic in an easily accessible case, namely – in the effects of multi blowings-up for curves in manifolds and for coherent sheaves on 2-dimensional manifolds. All the needed facts from analytic geometry can be find in the author’s books [L1], [L2]. 2. The canonical blowing-up of Cn at 0 The blow-up of Cn at 0 is Π = Πn = {(z, λ) : z ∈ λ} ⊂ Cn × P, P = Pn−1. Taking the inverse atlas for Cn × P γk : Cn × Cn−1 3 (z, w(k)) 7→ (z, C(w1, ..., 1 (k) , ..., wn)) ∈ Cn × {P \ P({zk = 0})) = Gk, k = 1, ..., n, 2010 Mathematics Subject Classification. Primary 32Sxx, Secondary 14Hxx. Key words and phrases. Resolution of singularities, curve, blowing-up, coherent analytic sheaf. ∗) This article was published (in Polish) in the proceedings of Xth Workshop on Theory of Extremal Problems (1989) and has never appeared in translation elsewhere. To honor this outstanding mathematician (who passed away in 2002) this article was translated into English (by T. Krasiński) in order to make it accesible to the mathematical community. ∗∗) The translator thanks Dinko Pervan (an Erasmus student from Croatia) for preparing the article in TeX and W. Kucharz, A. Płoski and Sz. Brzostowski for improving the English text. 11 12 STANISŁAW ŁOJASIEWICZ k (Π) = {(z, w(k)) : z ∈ C(w1, ..., 1, ..., wn)} =(cid:8)(z, w(k)) : z(k) = zkw(k) (cid:9) ; (that is γk = (id Cn)× (inverse mapping to the k-th canonical map on P)), we have the inverse images of Π Γk = γ−1 they are graphs of the polynomial mappings (zk, w(k)) → zkw(k), whence Π ⊂ Cn × P is an n-dimensional closed submanifold, (γk)Γk : Γk → Π ∩ Gk – its inverse maps (they give an inverse atlas on Π); composing them with biholomorphisms: (zk, w(k)) → (zkw1, ..., zk, ..., zkwn, w(k)) (domains onto the graphs of the preceding polynomial mappings) we obtain an inverse atlas on Π (∗) χk : Cn 3 (zk, w(k)) → (zkw1, ..., zk, ..., zkwn, C(w1, ..., 1, ..., wn)) ∈ Π ∩ Gk. The canonical projection p : Π → Cn is called the canonical blowing-up. The fiber S0 = p−1(0) = 0 × P (biholomorphic to P) is called the exceptional set (the exceptional submanifold); ΠCn\0 is the graph of the holomorphic mapping Cn \ 0 3 z → Cz ∈ P, whence p Cn\0 : ΠCn\0 → Cn \ 0 is a biholomorphism. Hence the blowing-up p : Π → Cn is a modification of Cn at 0. The inverse image p−1(E) of a set E ⊂ Cn in the k-th coordinate system (∗) can be expressed by (∗∗) In particular χ−1 (cid:26) p ◦ χk 3 (zk, w(k)) → (zkw1, ..., zk, ..., zkwn) ∈ Cn. k (S0) = {zk = 0}. k (p−1(E)) = (p ◦ χk)−1(E) where χ−1 The restrictions pΩ : ΠΩ → Ω, where Ω is an open neighbourhood of 0 at Cn, are called the local canonical blowings-up. 3. The blowing-up of a manifold at a point Let M be an n-dimensional manifold and a ∈ M. A blowing-up of M at the point a is a holomorphic mapping of manifolds π : ¯M → M such that πM\a : ¯M\π−1(a) → M\a is a biholomorphism and for an open neighbourhood U of a, the mapping πU is isomorphic to a local canonical blowing-up pΩ i.e. we have a commutative diagram π−1(U) ¯φ - p−1(Ω) πU ? U pΩ ? Ω φ - for some biholomorphisms φ : U → Ω, φ(a) = 0 and ¯φ : π−1(U) → p−1(Ω). (Notice that U and Ω can be abitrarily diminished). π is a proper mapping (because πM\a and πU are proper). The fiber S = π−1(a), biholomorphic to P, is called DESINGULARIZATION OF CURVES 13 the exceptional set (the exceptional submanifold) of the blowing-up. Thus π is a modification of M at a. φ−1 ◦ pΩ is a closed set in ΠΩ × M and(cid:0)φ−1 ◦ pΩ(cid:1) ∩ (ΠΩ × M\a) = φ−1 The existence of blowing-up. We take a chart (a coordinate system) at a: φ : U → Ω, φ(a) = 0, and define ¯M as a gluing-up of πΩ with M\a by the biholomor- phism (φU\a)−1 ◦ pΩ\0 : ΠΩ\0 → U\a. (Its graph is closed in ΠΩ × (M\a) because U\a ◦ pΩ\0). So we have the identifying biholomorphisms h0 : ΠΩ → D0, h1 : M\a → D1, where Di ⊂ ¯M , i = 0, 1, are open sets, ¯M = D0 ∪ D1 and h−1 U\a ◦ pΩ\0. Hence 1 (D0) = U\a (the domains of both sides) which implies h1(U\a) ⊂ D0. Next h−1 g = φ−1◦ p◦ h−1 : D0 → M contains (h−1 1 ∪g : ¯M → M is a holomorphic maping. Then πM\a = h−1 0 = gM\a) is a biholomorphism on the image. At last, φ◦ πU ⊃ φ◦ g ⊃ pΩ ◦ h−1 0 which implies 1 (U\a) ∪ D0 = D0), the equality, because the domains are equal (π−1(U) = h−1 whence the above diagram is commutative with ¯φ := h−1 0 . 1 ◦ h0 = φ−1 0 1 )D0 , and hence π = h−1 (because h−1 ⊃ φ−1 ◦ pΩ\0 ◦ h−1 1 Remark 1. Obviously, if G is an open neighbourhood of a at M then π : ¯M → M is a blowing-up at a if and only if πM\a is a biholomorphism and πG is a blowing-up at a. Proposition 1. If h : M → N is a biholomorphism of manifolds, h(a) = b, π1 : ¯M → M is a blowing-up at a, π2 : ¯N → N a blowing-up at b, then there exists a biholomorphism ¯h : ¯M → ¯N such that the diagram ¯M ? M π1 ¯h - ¯N π2 ? N h - (#) is commutative
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