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Euler Sequence
In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring
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Mathematics
Mathematics
Mathematics
(from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek out patterns[8][9] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist
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Exterior Power
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude[3] of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation—a choice of clockwise or counterclockwise. When regarded in this manner, the exterior product of two vectors is called a 2-blade
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Mathematical Reviews
Mathematical Reviews is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science.[1][2] The AMS also publishes an associated online bibliographic database called MathSciNet which contains an electronic version of Mathematical Reviews and additionally contains citation information for almost 3 million papers.Contents1 Reviews 2 Online database 3 Mathematical citation quotient 4 Current Mathematical Publications 5 See also 6 References 7 External linksReviews[edit] Mathematical Reviews was founded by Otto E
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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
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International Standard Book Number
"ISBN" redirects here. For other uses, see ISBN (other).International Standard Book
Book
NumberA 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar codeAcronym ISBNIntroduced 1970; 48 years ago (1970)Managing organisation International ISBN AgencyNo. of digits 13 (formerly 10)Check digit Weighted sumExample 978-3-16-148410-0Website www.isbn-international.orgThe International Standard Book
Book
Number (ISBN) is a unique[a][b] numeric commercial book identifier. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.[1] An ISBN is assigned to each edition and variation (except reprintings) of a book. For example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, and 10 digits long if assigned before 2007
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Graduate Texts In Mathematics
Graduate Texts in Mathematics
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
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
Mathematics
series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level.Contents1 List of books 2 See also 3 Notes 4 External linksList of books[edit]Introduction to Axiomatic Set Theory, Gaisi Takeuti, Wilson M
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Algebraic Geometry (book)
Algebraic Geometry is an influential,[1] algebraic geometry textbook written by Robin Hartshorne
Robin Hartshorne
and published by Springer-Verlag
Springer-Verlag
in 1977.Contents1 Importance 2 Contents 3 Notes 4 ReferencesImportance[edit] It was the first extended treatment of scheme theory written as a text intended to be accessible to graduate students. Contents[edit] The first chapter, titled "Varieties", deals with the classical algebraic geometry of varieties over algebraically closed fields. This chapter uses many classical results in commutative algebra, including Hilbert's Nullstellensatz, with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form the technical heart of the book
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Robin Hartshorne
Robin Cope Hartshorne (born March 15, 1938) is an American mathematician. Hartshorne is an algebraic geometer who studied with Oscar Zariski, David Mumford, Jean-Pierre Serre
Jean-Pierre Serre
and Alexander Grothendieck. He was a Putnam Fellow in Fall, 1958. He received his doctorate from Princeton University
Princeton University
in 1963[1] and then became a Junior Fellow at Harvard University, where he taught for several years. In the 1970s he was appointed to the faculty at the University of California, Berkeley. He is currently[when?] retired. Hartshorne is the author of the text Algebraic Geometry. In 2012 he became a fellow of the American Mathematical Society.[2]Contents1 Selected publications 2 See also 3 References 4 External linksSelected publications[edit]Foundations of Projective Geometry, New York: W. A. Benjamin, 1967; Algebraic Geometry, New York: Springer-Verlag, 1977;[3] corrected 6th printing, 1993
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Geometric Genus
In algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties and complex manifolds.Contents1 Definition 2 Case of curves 3 Genus of singular varieties 4 See also 5 Notes 6 ReferencesDefinition[edit] The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number hn,0 (equal to h0,n by Serre duality), that is, the dimension of the canonical linear system plus one. In other words for a variety V of complex dimension n it is the number of linearly independent holomorphic n-forms to be found on V.[1] This definition, as the dimension ofH0(V,Ωn)then carries over to any base field, when Ω is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power, the canonical line bundle. The geometric genus is the first invariant pg = P1 of a sequence of invariants Pn called the plurigenera. Case of curves[edit] In the c
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Ample Line Bundle
In algebraic geometry, a very ample line bundle is one with enough global sections to set up an embedding of its base variety or manifold M displaystyle M into projective space. An ample line bundle is one such that some positive power is very ample
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Fano Varieties
In algebraic geometry, a Fano variety, introduced in (Fano 1934, 1942), is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities.Contents1 Examples 2 Some properties 3 Classification in small dimensions 4 Notes 5 References 6 See alsoExamples[edit]The fundamental example of Fano varieties are the projective spaces: the anticanonical line bundle of Pn over a field k is O(n+1), which is very ample (over the complex numbers, its curvature is n+1 times the Fubini–Study symplectic form). Let D be a smooth codimension-1 subvariety in Pn. The adjunction formula implies that KD = (KX + D)D = (−(n+1)H + deg(D)H)D, where H is the class of a hyperplane
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Canonical Sheaf
In mathematics, the canonical bundle of a non-singular algebraic variety V displaystyle V of dimension n displaystyle n over a field is the line bundle Ω n = ω displaystyle ,!Omega ^ n =omega , which is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V. This is the dualising object for Serre duality on V. It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor
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Vector Field
In vector calculus, a vector field is an assignment of a vector to each point in a subset of space.[1] A vector field in the plane (for instance), can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. The elements of differential and integral calculus extend naturally to vector fields
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Exact Sequence
An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry
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Vector Space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a vector space. They represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector
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