# 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 record at the present prestige of vital learn themes in algebraic geometry and its purposes, similar to computational algebra and geometry, singularity idea algorithms, numerical strategies of polynomial structures, coding conception, communique networks, and desktop imaginative and prescient. Contributions on extra basic facets of algebraic geometry comprise expositions regarding counting issues on kinds over finite fields, Mori thought, linear platforms, Abelian forms, vector bundles on singular curves, degenerations of surfaces, and replicate symmetry of Calabi-Yau manifolds.

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P~lll) of plane curves. a (pi;! • , p;:lll) is special if and only if it is ( -1) -special. 4. Segre Implies Harbourne-Hirschowitz Note that Harbourne-Hirschowitz Conjecture immediately implies Segre's Conjecture, since (- 1)-special systems are not reduced. In this section we will show that in fact Segre's Conjecture is not weaker that the Harbourne-Hirschowitz. We keep the notation from the previous section. a(p~1 , ... ,p:"). 1. Suppose that Segre:S' Conjecture is true. Let PI, "',Pn be general points in the plane and let S be the blow-up of the plane at PI, ...

E. each C is an elliptic curve. 3, since the C's are disjoint. Therefore we must have k = 1 and we set C I = C, hi = h. Let F be an irreducible component of M; since F . 3 again and conclude that F has genus zero and dim IFI = O. This gives a contradiction, since F moves in its linear system. We conclude that M = O. i = 1.... 3 to C and D j and conclude that each D j is rational. 4,(i) implies that it is a (-1 )-curve (since D] = D j · Ks and the sum is -2 by rationality). The assertion about the non-speciality of the system is an easy computation: v(hlC I + DI + ...

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.