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Intersections of Loci on an Algebraic V4

  • J. A. Todd (a1)

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The problem of residual intersections is one of considerable importance in algebraic geometry. In its most general form the problem may be stated as follows. On a given variety V of d dimensions two varieties A and B, of respective dimensions k and k′, pass through a variety C whose dimension r is not less than r′ = k + k′ – d. It is required to determine the variety D of dimension r′ which forms the residual intersection of A and B. The classical paper on this subject is that of Severi*. He considers the case in which V is a linear space, and obtains a large variety of enumerative results connecting the characters of the residual intersection with those of the given loci.

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* Severi, F., Mem. di Torino, (2), 52 (1903), 61; see also the account in the Encyk. Mat. Wiss. III, C. 7, 944.

Segre, B., Mem. R. Acc. d'Italia, 5 (1934), 479.

For an account of these see Baker, H. F., Principles of geometry, vol. 6 (Cambridge, 1933), Ch. vi.

§ See Severi, F., Mem. R. Acc. d'ltalia, 4 (1932), 71; Todd, J. A., Annals of Math. 35 (1934), 702.

* The dimension of any or all of the systems may be zero.

Todd, J. A., Proc. London Math. Soc. (2), 43 (1937), 127.

* Or primals of some higher order.

* If | S | is a net of surfaces on A, and if J is the Jacobian curve of the net, then YA ≡- J – 3(SX A) – 6(S 2). The nodes of A are evidently base-points of the series of equivalence {J}

* See, e.g., the chapter in Baker's volume, previously quoted.

Todd, J. A., Proc. London Math. Soc. (2), 43 (1937), 195.

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