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Exterior algebra methods for the construction of rational surfaces in the projective fourspace. (English) Zbl 1107.14039
Lossen, Christoph (ed.) et al., Singularities and computer algebra. Selected papers of the conference, Kaiserslautern, Germany, October 18–20, 2004 on the occasion of Gert-Martin Greuel’s 60th birthday. Cambridge: Cambridge University Press (ISBN 0-521-68309-2/pbk). London Mathematical Society Lecture Note Series 324, 1-12 (2006).
The authors present a computer-aided method for producing examples of smooth rational surfaces in the projective space $${\mathbb P}^4$$ over the complex field. The starting point is the monad construction of Beilinson, which provides examples of such surfaces, once one can show the existence of maps between differential sheaves, with some preassigned properties. In some cases, the authors are able to construct the maps, with the aid of a computer algebra package, over a field of positive characteristic. They prove that the resulting surface $$X_0$$ is rational, and determine its plane model. Then they prove that $$X_0$$ deforms to a surface defined over a number field, finitely generated over $$\mathbb Q$$.
With this method, the authors construct a family of smooth rational surfaces in $${\mathbb P}^4$$, isomorphic to the blow up of $${\mathbb P}^2$$ at $$21$$ points in very special position.
For the entire collection see [Zbl 1086.14001].
##### MSC:
 14M07 Low codimension problems in algebraic geometry 14J26 Rational and ruled surfaces 14Q10 Computational aspects of algebraic surfaces
surfaces
##### Software:
Macaulay2; SINGULAR
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