The Geometry of Schemes by Joe Harris & David Eisenbud

Author:Joe Harris & David Eisenbud
Language: eng
Format: epub
Publisher: Springer New York
Published: 2014-07-10T00:00:00+00:00

Computing the Hilbert function of S/I from this resolution, we see that it is the same as that of S/I(X), and since I ⊂ I(X), we must have I = I(X); that is, I(X) is generated by q1 and q2 and X is correspondingly the intersection of the two conics containing it.

Summing up, we see that all three of the examples look the same from the point of view of Hilbert polynomials; the first two examples are distinguished by their Hilbert functions; and the last two examples look the same from the point of view of Hilbert functions but are distinguished by their graded Betti numbers. It is not hard to find corresponding examples of subschemes X of length 4 where the properties distinguished are actually intrinsic properties of the schemes, not dependent on the embedding. For example, while the scheme Spec K[x]/(x4) may be embedded in so as to have any of the Hilbert functions and Betti numbers above (for instance, as the subschemes defined by the ideals , and respectively, the subscheme defined by will always have the graded Betti numbers and Hilbert function of case 3).

Exercise III-63. Find the Hilbert polynomial, the Hilbert function, and the graded Betti numbers of all subschemes of the plane of length 3.

Examples: Double Lines in General and in . So far, most of our discussion of projective schemes has been parallel to the theory of varieties. We will now look at one genuinely nonclassical family of examples.

Exercise II-35 asked you to show that all affine double lines are equivalent. This is not true for projective double lines. Here are some simple examples.

Let K be a field. Consider the graded ring

and the scheme

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