Download Applications of Algebraic Geometry to Coding Theory, Physics by I. Burban, Yu. Drozd, G.-M Greuel (auth.), Ciro Ciliberto, PDF

By I. Burban, Yu. Drozd, G.-M Greuel (auth.), Ciro Ciliberto, Friedrich Hirzebruch, Rick Miranda, Mina Teicher (eds.)

An up to date file at the present prestige of vital learn themes in algebraic geometry and its functions, akin to computational algebra and geometry, singularity concept algorithms, numerical suggestions of polynomial platforms, coding conception, verbal exchange networks, and laptop imaginative and prescient. Contributions on extra basic facets of algebraic geometry comprise expositions on the topic of counting issues on types over finite fields, Mori thought, linear platforms, Abelian types, vector bundles on singular curves, degenerations of surfaces, and replicate symmetry of Calabi-Yau manifolds.

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Additional info for Applications of Algebraic Geometry to Coding Theory, Physics and Computation

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Recall that the set E(K) of the rational points of an elliptic curve over K = Q can be structured as an abelian algebraic group that is always finitely generated by a theorem of Mordell, see for example (Silverman). In fact, it is E(Q) = E(Q)tors EB Z', where r is called rank of the curve. Since the number of rational points with finite order is at most 16 (Mazur; Cohen, Th. 11), the curve has infinite rational points if, and only if, its rank is at least 1. The question of deciding whether the curve has positive rank is still open.

2); hence if it holds for one curve, then it holds also for the general curve in the same irreducible component. The maximal rank for general curves has been deeply studied by E. Ballico and P. Ellia who proved that a general non degenerate and non special curve of genus g 2': 0 and degree d 2': g + 11 in IP'I1 has maximal rank, where 11 2': 3 (see (Ballico and Ellia) and the references therein). In (Orecchia) disjoint unions ofrational curves with maximal 23 C. Ciliberto et af. ). Applications of Algebraic Geometry to Coding Theory.

Mn be positive integers. We let ~J be the linear system of plane curves of degree d and ~ (p';ll .... , p;;lll) the sub-linear system of ~ of curves having multiplicity at least mj at the point Pi, i = 1, ... , n. ( /111 11111) Pi! ~PI ""'P/1' Let 1t : S -+ JlD2 be the blow-up of JlD2 at the points PI, ... ,Pn. Let £ be a line bundle on S, or, by abusing notation, the corresponding complete linear system. One defines the virtual dimension of £ to be: v(£) := X(£) - I = £. (£-Ks) 2 where Ks is the canonical class on S.

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