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By Otto Mutzbauer

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146. 2. Alf Aure, Wolfram Decker, Klaus Hulek, Sorin Popescu, and Kristian Ranestad: Syzygies of abelian and bielliptic surfaces in p4. Internat. J. , 8(7):849919, 1997. 3. Wolfram Decker and Frank-Olaf Schreyer: Non-general type surfaces in p4: some remarks on bounds and constructions. J. , 29(4-5):545582,2000. Symbolic computation in algebra, analysis, and geometry (Berkeley, CA,1998). 4. David Eisenbud: Commutative algebra. Springer-Verlag, New York, 1995. With a view toward algebraic geometry.

This would make further computations less efficient. Thus we take a different approach: since the cotangent sequence given in the previous section is a sequence of locally free sheaves, it is locally split, and thus remains exact when tensored by Ox. Consequently illl'n Q9 Ox is also represented by the kernel of the map f Q9 Ox, where f is the map used in the definition of the cotangent bundle of IP"'. ) Since the map is a map between free modules over ringP3/idealCubic, the kernel has depth (at least) two.

Perhaps the most basic invariant of X is its dimension: i55 : dim X 055 = 1 This shows that X consists of a curve, and possibly some zero-dimensional components. The command i56 : idealXtop = top idealX 2 2 2 2 - 2x x x - x x x + x X 1 013 123 3 4 056 ideal (x 056 Ideal of ringP3 ° J 2 2 2 x - 10915x x x - 10917x 012 1 X ° 30 D. Eisenbud returns the ideal of the largest dimensional components of X. If there were O-dimensional components (or if idealX were not saturated) then idealXtop would be larger than idealX.

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