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Clifford's Theorem On Special Divisors
In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve ''C''. Statement A divisor on a Riemann surface ''C'' is a formal sum \textstyle D = \sum_P m_P P of points ''P'' on ''C'' with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of ''C,'' defining L(D) as the vector space of functions having poles only at points of ''D'' with positive coefficient, ''at most as bad'' as the coefficient indicates, and having zeros at points of ''D'' with negative coefficient, with ''at least'' that multiplicity. The dimension of L(D) is finite, and denoted \ell(D). The linear system of divisors attached to ''D'' is the corresponding projective space of dimension \ell(D)-1. The other significant invariant of ''D'' is its degree ''d'', which is the sum of all its coefficients. A divisor is called ''special'' if ''ℓ''(''K'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gonality
In mathematics, the gonality of an algebraic curve ''C'' is defined as the lowest degree of a nonconstant rational map from ''C'' to the projective line. In more algebraic terms, if ''C'' is defined over the field ''K'' and ''K''(''C'') denotes the function field of ''C'', then the gonality is the minimum value taken by the degrees of field extensions :''K''(''C'')/''K''(''f'') of the function field over its subfields generated by single functions ''f''. If ''K'' is algebraically closed, then the gonality is 1 precisely for curves of genus 0. The gonality is 2 for curves of genus 1 (elliptic curves) and for hyperelliptic curves (this includes all curves of genus 2). For genus ''g'' ≥ 3 it is no longer the case that the genus determines the gonality. The gonality of the generic curve of genus ''g'' is the floor function of :(''g'' + 3)/2. Trigonal curves are those with gonality 3, and this case gave rise to the name in general. Trigonal curves include the Picard curves, of gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curves
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philosophical Transactions Of The Royal Society Of London
''Philosophical Transactions of the Royal Society'' is a scientific journal published by the Royal Society. In its earliest days, it was a private venture of the Royal Society's secretary. It was established in 1665, making it the first journal in the world exclusively devoted to science, and therefore also the world's longest-running scientific journal. It became an official society publication in 1752. The use of the word ''philosophical'' in the title refers to natural philosophy, which was the equivalent of what would now be generally called ''science''. Current publication In 1887 the journal expanded and divided into two separate publications, one serving the physical sciences ('' Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences'') and the other focusing on the life sciences ('' Philosophical Transactions of the Royal Society B: Biological Sciences''). Both journals now publish themed issues and issues resulting from pap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruth Lyttle Satter Prize In Mathematics
The Ruth Lyttle Satter Prize in Mathematics, also called the Satter Prize, is one of twenty-one prizes given out by the American Mathematical Society (AMS). It is presented biennially in recognition of an outstanding contribution to mathematics research by a woman in the previous six years. The award was established in 1990 using a donation from Joan Birman, in memory of her sister, Ruth Lyttle Satter, who worked primarily in biological sciences, and was a proponent for equal opportunities for women in science. First awarded in 1991, the award is intended to "honor atter'scommitment to research and to encourage women in science". The winner is selected by the council of the AMS, based on the recommendation of a selection committee. The prize is awarded at the Joint Mathematics Meetings during odd numbered years, and has always carried a modest cash reward. Since 2003, the prize has been $5,000, while from 1997 to 2001, the prize came with $1,200, and prior to that it was $4,00 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claire Voisin
Claire Voisin (born 4 March 1962) is a French mathematician known for her work in algebraic geometry. She is a member of the French Academy of Sciences and holds the chair of Algebraic Geometry at the Collège de France. Work She is noted for her work in algebraic geometry particularly as it pertains to variations of Hodge structures and mirror symmetry, and has written several books on Hodge theory. In 2002, Voisin proved that the generalization of the Hodge conjecture for compact Kähler varieties is false. The Hodge conjecture is one of the seven Clay Mathematics Institute Millennium Prize Problems which were selected in 2000, each having a prize of one million US dollars. Voisin won the European Mathematical Society Prize in 1992 and the Servant Prize awarded by the Academy of Sciences in 1996. She received the Sophie Germain Prize in 2003 and the Clay Research Award in 2008 for her disproof of the Kodaira conjecture on deformations of compact Kähler manifolds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Lazarsfeld
Robert Kendall Lazarsfeld (born April 15, 1953) is an American mathematician, currently a professor at Stony Brook University. He was previously the Raymond L. Wilder Collegiate Professor of Mathematics at the University of Michigan. He is the son of two sociologists, Paul Lazarsfeld and Patricia Kendall. His research focuses on algebraic geometry. During 2002–2009, Lazarsfeld was an editor at the '' Journal of the American Mathematical Society'' (Managing Editor, 2007–2009). In 2012–2013, he served as the Managing Editor of the ''Michigan Mathematical Journal''. Lazarsfeld went to Harvard for undergraduate studies and earned his doctorate from Brown University in 1980 under supervision of William Fulton. In 2006 Lazarsfeld was elected a Fellow of the American Academy of Arts and Sciences. In 2012 he became a fellow of the American Mathematical Society. In 2015 he was awarded the AMS Leroy P. Steele Prize The Leroy P. Steele Prizes are awarded every year by the A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Betti Number
In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X''''N''thinsp;/''I'' where ''I'' is the homogeneous ideal defining ''V'', ''K'' is the algebraically closed field over which ''V'' is defined, and :''K'' 'X''0, ''X''1, ''X''2, ..., ''X''''N'' is the polynomial ring in ''N'' + 1 variables ''X''''i''. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra. Formulation Since ''V'' is assumed to be a variety, and so an irreducible algebraic set, the ideal ''I'' can be chosen to be a prime ideal, and so ''R'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Coordinate Ring
In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X''''N''thinsp;/''I'' where ''I'' is the homogeneous ideal defining ''V'', ''K'' is the algebraically closed field over which ''V'' is defined, and :''K'' 'X''0, ''X''1, ''X''2, ..., ''X''''N'' is the polynomial ring in ''N'' + 1 variables ''X''''i''. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra. Formulation Since ''V'' is assumed to be a variety, and so an irreducible algebraic set, the ideal ''I'' can be chosen to be a prime ideal, and so ''R'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Resolution
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the zero-object. Generally, the objects in the sequence are restricted to have some property ''P'' (for example to be free). Thus one speaks of a ''P resolution''. In particular, every module has free resolutions, projective resol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
