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Workshop on geometry in algebra and algebra in geometry
São Paulo Journal of Mathematical Sciences, 2021This short note describes some of the contributions to the Workshop GAAG 2019 held in Medellin, Colombia.
H. Bursztyn +4 more
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The Geometry of Cubic Hypersurfaces
, 2023Cubic hypersurfaces are described by almost the simplest possible polynomial equations, yet their behaviour is rich enough to demonstrate many of the central challenges in algebraic geometry. With exercises and detailed references to the wider literature,
D. Huybrechts
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The Algebra-Geometry Dictionary
1992In this chapter, we will explore the correspondence between ideals and varieties. In §§1 and 2, we will prove the Nullstellensatz, a celebrated theorem which identifies exactly which ideals correspond to varieties. This will allow us to construct a “dictionary” between geometry and algebra, whereby any statement about varieties can be translated into a
Donal O’Shea +2 more
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Geometry, Algebra, and Algorithms
1992This chapter will introduce some of the basic themes of the book. The geometry we are interested in concerns affine varieties, which are curves and surfaces (and higher dimensional objects) defined by polynomial equations. To understand affine varieties, we will need some algebra, and in particular, we will need to study ideals in the polynomial ring k[
Donal O’Shea +2 more
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1992
So far all of the varieties we have studied have been subsets of affine space kn. In this chapter, we will enlarge kn by adding certain “points at ∞” to create n-dimensional projective space \(\mathbb{P}^{n}(k)\). We will then define projective varieties in \(\mathbb{P}^{n}(k)\) and study the projective version of the algebra–geometry dictionary.
David A. Cox +2 more
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So far all of the varieties we have studied have been subsets of affine space kn. In this chapter, we will enlarge kn by adding certain “points at ∞” to create n-dimensional projective space \(\mathbb{P}^{n}(k)\). We will then define projective varieties in \(\mathbb{P}^{n}(k)\) and study the projective version of the algebra–geometry dictionary.
David A. Cox +2 more
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, 1994 These notes follow a first course in algebraic geometry designed for second year graduate students at the University of Michigan. The recommended texts accompanying this course include Basic Algebriac Geometry I by Igor R. Shafarevich, Algebraic Geometry, A First Course by Joe Harris, An Invitation to Algebraic Geometry by Karen Smith, and Algebraic ...
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Algebraic Geometry and Arithmetic Curves
, 2002Introduction 1. Some topics in commutative algebra 2. General Properties of schemes 3. Morphisms and base change 4. Some local properties 5. Coherent sheaves and Cech cohmology 6. Sheaves of differentials 7.
Qing Liu
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Improved decoding of Reed-Solomon and algebraic-geometry codes
IEEE Transactions on Information Theory, 1999Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string.
V. Guruswami, M. Sudan
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Algebraic Geometry for Associative Algebras
2000The noncommutative site structure sheaves and their sections regular algebras valuations and divisors cohomology theories a functorial approach formalizing the topology.
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