We investigate properties of the contact exponent (in the sense of Hironaka [Hi]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic. We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [GB-P1].
{"title":"Contact exponent and the Milnor number of plane curve singularities","authors":"E. G. Barroso, A. Płoski","doi":"10.18778/8142-814-9.08","DOIUrl":"https://doi.org/10.18778/8142-814-9.08","url":null,"abstract":"We investigate properties of the contact exponent (in the sense of Hironaka [Hi]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic. We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [GB-P1].","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"138 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116392850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Dumnicki, L. Farnik, Krishna Hanumanthu, G. Malara, T. Szemberg, J. Szpond, H. Tutaj-Gasinska
We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an elliptic curve and nine Fermat points. As a consequence of our analysis, we also show that the Bounded Negativity Conjecture holds for the surfaces we consider. The note contains also some problems for future attention.
{"title":"Negative curves on special rational surfaces","authors":"M. Dumnicki, L. Farnik, Krishna Hanumanthu, G. Malara, T. Szemberg, J. Szpond, H. Tutaj-Gasinska","doi":"10.18778/8142-814-9.06","DOIUrl":"https://doi.org/10.18778/8142-814-9.06","url":null,"abstract":"We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an elliptic curve and nine Fermat points. As a consequence of our analysis, we also show that the Bounded Negativity Conjecture holds for the surfaces we consider. The note contains also some problems for future attention.","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127479325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present note we study some extreme properties of point-line configurations in the complex projective plane from a viewpoint of algebraic geometry. Using Hirzebruch-type inequalites we provide some new results on r-rich lines, symplicial arrangements of lines, and the so-called free line arrangmenets.
{"title":"Extremal properties of line arrangements in the complex projective plane","authors":"Piotr Pokora","doi":"10.18778/8142-814-9.14","DOIUrl":"https://doi.org/10.18778/8142-814-9.14","url":null,"abstract":"In the present note we study some extreme properties of point-line configurations in the complex projective plane from a viewpoint of algebraic geometry. Using Hirzebruch-type inequalites we provide some new results on r-rich lines, symplicial arrangements of lines, and the so-called free line arrangmenets.","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120947454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a survey of the research on rings of polynomial constants and fields of rational constants of cyclic factorizable derivations in polynomial rings over fields of characteristic zero. 1. Motivations and preliminaries The first inspiration for the presented series of articles (some of them are joint works with Hegedűs and Ossowski) was the publication [20] of professor Nowicki and professor Moulin Ollagnier. The fundamental problem investigated in that series of articles concerns rings of polynomial constants ([26], [28], [33], [29], [8]) and fields of rational constants ([30], [31], [32]) in various classes of cyclic factorizable derivations. Moreover, we investigate Darboux polynomials of such derivations together with their cofactors ([33]) and applications of the results obtained for cyclic factorizable derivations to monomial derivations ([31]). Let k be a field. If R is a commutative k-algebra, then k-linear mapping d : R→ R is called a k-derivation (or simply a derivation) of R if d(ab) = ad(b) + bd(a) for all a, b ∈ R. The set R = ker d is called a ring (or an algebra) of constants of the derivation d. Then k ⊆ R and a nontrivial constant of the derivation d is an element of the set R k. By k[X] we denote k[x1, . . . , xn], the polynomial ring in n variables. If f1, . . . , fn ∈ k[X], then there exists exactly one derivation d : k[X]→ k[X] such that d(x1) = f1, . . . , d(xn) = fn. 2010 Mathematics Subject Classification. 13N15, 12H05, 34A34.
{"title":"Rings and fields of constants of cyclic factorizable derivations","authors":"J. Zieliński","doi":"10.18778/8142-814-9.16","DOIUrl":"https://doi.org/10.18778/8142-814-9.16","url":null,"abstract":"We present a survey of the research on rings of polynomial constants and fields of rational constants of cyclic factorizable derivations in polynomial rings over fields of characteristic zero. 1. Motivations and preliminaries The first inspiration for the presented series of articles (some of them are joint works with Hegedűs and Ossowski) was the publication [20] of professor Nowicki and professor Moulin Ollagnier. The fundamental problem investigated in that series of articles concerns rings of polynomial constants ([26], [28], [33], [29], [8]) and fields of rational constants ([30], [31], [32]) in various classes of cyclic factorizable derivations. Moreover, we investigate Darboux polynomials of such derivations together with their cofactors ([33]) and applications of the results obtained for cyclic factorizable derivations to monomial derivations ([31]). Let k be a field. If R is a commutative k-algebra, then k-linear mapping d : R→ R is called a k-derivation (or simply a derivation) of R if d(ab) = ad(b) + bd(a) for all a, b ∈ R. The set R = ker d is called a ring (or an algebra) of constants of the derivation d. Then k ⊆ R and a nontrivial constant of the derivation d is an element of the set R k. By k[X] we denote k[x1, . . . , xn], the polynomial ring in n variables. If f1, . . . , fn ∈ k[X], then there exists exactly one derivation d : k[X]→ k[X] such that d(x1) = f1, . . . , d(xn) = fn. 2010 Mathematics Subject Classification. 13N15, 12H05, 34A34.","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130076679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this survey we present recent results in the study of the medial axes of sets definable in polynomially bounded o-minimal structures. We take the novel point of view of singularity theory. Indeed, it has been observed only recently that the medial axis — i.e. the set of points with more than one closest point to a given closed set X ⊂ R n (with respect to the Euclidean distance) — reaches some singular points of X bringing along some metric information about them.
{"title":"When the medial axis meets the singularities","authors":"M. Denkowski","doi":"10.18778/8142-814-9.05","DOIUrl":"https://doi.org/10.18778/8142-814-9.05","url":null,"abstract":". In this survey we present recent results in the study of the medial axes of sets definable in polynomially bounded o-minimal structures. We take the novel point of view of singularity theory. Indeed, it has been observed only recently that the medial axis — i.e. the set of points with more than one closest point to a given closed set X ⊂ R n (with respect to the Euclidean distance) — reaches some singular points of X bringing along some metric information about them.","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114355142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aleksandra Gala-Jaskórzynska, K. Kurdyka, K. Rudnicka, S. Spodzieja
We give a bound of the height of a multipolynomial resultant in terms of polynomial degrees, the resultant of which applies. Additionally we give a Gelfond-Mahler type bound of the height of homogeneous divisors of a homogeneous polynomial.
{"title":"Gelfond-Mahler inequality for multipolynomial resultants","authors":"Aleksandra Gala-Jaskórzynska, K. Kurdyka, K. Rudnicka, S. Spodzieja","doi":"10.18778/8142-814-9.07","DOIUrl":"https://doi.org/10.18778/8142-814-9.07","url":null,"abstract":"We give a bound of the height of a multipolynomial resultant in terms of polynomial degrees, the resultant of which applies. Additionally we give a Gelfond-Mahler type bound of the height of homogeneous divisors of a homogeneous polynomial.","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126863906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the dual Hesse arrangement","authors":"M. Lampa-Baczyńska, D. Wójcik","doi":"10.18778/8142-814-9.12","DOIUrl":"https://doi.org/10.18778/8142-814-9.12","url":null,"abstract":"","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126451496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We prove that in order to find the value of the Łojasiewicz exponent ł ( f ) of a Kouchnirenko non-degenerate holomorphic function f : ( C n , 0) → ( C , 0) with an isolated singular point at the origin, it is enough to find this value for any other (possibly simpler) function g : ( C n , 0) → ( C , 0) , provided this function is also Kouchnirenko non-degenerate and has the same Newton diagram as f does. We also state a more general problem, and then reduce it to a Teissier-like result on (c)-cosecant deformations, for formal power series with coefficients in an algebraically closed field K .
。我们证明为了找到的价值Łojasiewicz指数ł(f) Kouchnirenko不易变质的全纯函数f: (C n, 0)→(C, 0)与孤立奇点在原点,它足以找到这个值为任何其他函数g(可能更简单):(C n, 0)→(C, 0),提供这个功能也是Kouchnirenko简一样,牛顿图f。我们还陈述了一个更一般的问题，然后将其简化为(c)-余割变形上的Teissier-like结果，用于代数闭域K中带系数的形式幂级数。
{"title":"A note on the Łojasiewicz exponent of non-degenerate isolated hypersurface singularities","authors":"S. Brzostowski","doi":"10.18778/8142-814-9.04","DOIUrl":"https://doi.org/10.18778/8142-814-9.04","url":null,"abstract":". We prove that in order to find the value of the Łojasiewicz exponent ł ( f ) of a Kouchnirenko non-degenerate holomorphic function f : ( C n , 0) → ( C , 0) with an isolated singular point at the origin, it is enough to find this value for any other (possibly simpler) function g : ( C n , 0) → ( C , 0) , provided this function is also Kouchnirenko non-degenerate and has the same Newton diagram as f does. We also state a more general problem, and then reduce it to a Teissier-like result on (c)-cosecant deformations, for formal power series with coefficients in an algebraically closed field K .","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"145 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124148837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A non-containment example on lines and a smooth curve of genus 10","authors":"Marek Janasz, Grzegorz Malara","doi":"10.18778/8142-814-9.09","DOIUrl":"https://doi.org/10.18778/8142-814-9.09","url":null,"abstract":"","PeriodicalId":273656,"journal":{"name":"Analytic and Algebraic Geometry 3","volume":"200 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116498110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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