Segre classes on smooth projective toric varieties
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Mathematische Zeitschrift
Segre classes on smooth projective toric varieties Torgunn Karoline Moe · Nikolay Qviller
Received: 7 May 2012 / Accepted: 23 December 2012 © Springer-Verlag Berlin Heidelberg 2013
Abstract We provide a generalization of the algorithm of Eklund, Jost and Peterson for computing Segre classes of closed subschemes of projective k-space. The algorithm is here generalized to computing the Segre classes of closed subschemes of smooth projective toric varieties. Keywords Segre classes · Toric varieties · Computational algorithm · Nef cone · Intersection theory. Mathematics Subject Classification (2000) Secondary 14C20 · 14Q99
Primary 14C17 · 14M25;
1 Introduction 1.1 Background Segre classes are important objects appearing in intersection theory. Indeed, problems in for instance enumerative geometry frequently reduce to the computation of Segre classes. Nevertheless, the computation of the Segre class s(Z , X ) of a closed subscheme Z of a scheme X , from the raw information contained in the sheaf of ideals I Z alone, is a difficult problem. In terms of intersection theory, s(Z , X ) is a rational equivalence class of cycles supported on the scheme Z , but it is often sufficient simply to know the push-forward j∗ s(Z , X ) ∈ A∗ (X ), where j : Z → X denotes the inclusion map. If Z has dimension n, this push-forward has n + 1 components si ∈ An−i (X ), for 0 ≤ i ≤ n (some readers may react to a slightly unconventional index notation; we give an explanation in Sect. 1.3).
T. K. Moe (B) · N. Qviller Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway e-mail: [email protected] N. Qviller e-mail: [email protected]
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T. K. Moe, N. Qviller
Recently, Eklund et al. [5] gave an algorithm computing these components si , when Z is a closed subscheme of projective k-space over a field K, with only input the ideal of Z in the homogeneous coordinate ring K[x0 , . . . , xk ]. A related, but slightly different algorithm for computing Segre classes of subschemes of projective spaces had already been provided by Aluffi (see [1, Section 3]). The algorithm of Eklund–Jost–Peterson is based on using the theory of residual intersections, by choosing a set of generic schemes envelopping the scheme Z , and then considering (and computing the degree of) the residual schemes of Z in these. The result is a generic set of n + 1 linear equations in the si , which is obviously enough to provide these classes. The method is then turned into a probabilistic computer algorithm by replacing the term “generic” with the term “random.” Our aim is to generalize the algorithm mentioned above to the case of ambient smooth projective toric varieties. As mentioned, Segre classes are complicated objects, but for subschemes of projective spaces they are quite manageable, mainly because A∗ (Pk ) is simply Z[H ]/(H k+1 ), and also because these subschemes are easily described by homogeneous ideals in the coordinate ring. Now, Pk is a special instance of a k-dimensional toric variety, i.e.
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