Computing the equations of a variety
HTML articles powered by AMS MathViewer
- by Michela Brundu and Mike Stillman PDF
- Trans. Amer. Math. Soc. 337 (1993), 677-690 Request permission
Abstract:
Let $X \subset {\mathbb {P}^n}$ be a projective variety or subscheme, and let $\mathcal {F}$ be an invertible sheaf on $X$. A set of global sections of $\mathcal {F}$ determines a map from a Zariski open subset of $X$ to ${\mathbb {P}^r}$. The purpose of this paper is to find, given $X$ and $\mathcal {F}$, the homogeneous ideal defining the image in ${\mathbb {P}^r}$ of this rational map. We present algorithms to compute the ideal of the image. These algorithms can be implemented using only the computation of Gröbner bases and syzygies, and they have been implemented in our computer algebra system Macaulay. Our methods generalize to include the case when $X$ is an arbitrary projective scheme and $\mathcal {F}$ is generically invertible.References
-
D. Bayer, The division algorithm and the Hilbert scheme, Ph.D. thesis, Harvard University, 1982.
D. Bayer and M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra, Source and object code available for Unix and Macintosh computers. Contact the authors, or ftp zariski.harvard.edu, Name: ftp, Password: any, cd Macaulay, binary, get M3.tar, quit, tar xf M3.tar.
B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal, Ph.D. thesis, Universität Innsbruck, 1965.
—, Gröbner bases: An algorithmic method in polynomial ideal theory, Multidimensional Systems Theory (N. K. Bose, ed.), Reidel, 1985, pp. 184-232.
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1 D. Eisenbud and M. Stillman, Methods for computing in algebraic geometry and commutative algebra, in preparation.
- William Fulton, Algebraic curves. An introduction to algebraic geometry, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss. MR 0313252
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278 (French). MR 68874, DOI 10.2307/1969915
- J. Herzog, A. Simis, and W. V. Vasconcelos, Koszul homology and blowing-up rings, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 79–169. MR 686942
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 337 (1993), 677-690
- MSC: Primary 13P10; Secondary 13A30, 13D45, 14B15
- DOI: https://doi.org/10.1090/S0002-9947-1993-1091704-X
- MathSciNet review: 1091704