# zbMATH — the first resource for mathematics

A relative Segre zeta function. (English) Zbl 1453.14020
Summary: The choice of a homogeneous ideal in a polynomial ring defines a closed subscheme $$Z$$ in a projective space as well as an infinite sequence of cones over $$Z$$ in progressively higher dimension projective spaces. Recent work of Aluffi introduces the Segre zeta function, a rational power series with integer coefficients which captures the relationship between the Segre class of $$Z$$ and those of its cones. The goal of this note is to define a relative version of this construction for closed subschemes of projective bundles over a smooth variety. If $$Z$$ is a closed subscheme of such a projective bundle $$\mathbb{P}(E)$$, this relative Segre zeta function will be a rational power series which describes the Segre class of the cone over $$Z$$ in every projective bundle “dominating” $$\mathbb{P}(E)$$. When the base variety is a point we recover the absolute Segre zeta function for projective spaces. Part of our construction requires $$Z$$ to be the zero scheme of a section of a bundle on $$\mathbb{P}(E)$$ of rank smaller than that of $$E$$ that is able to extend to larger projective bundles. The question of what bundles may extend in this sense seems independently interesting and we discuss some related results, showing that at a minimum one can always count on direct sums of line bundles to extend. Furthermore, the relative Segre zeta function depends only on the Segre class of $$Z$$ and the total Chern class of the bundle defining $$Z$$, and the basic forms of the numerator and denominator can be described. As an application of our work we derive a Segre zeta function for products of projective spaces and prove its key properties.
##### MSC:
 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
Full Text:
##### References:
 [1] Aluffi, P., Computing characteristic classes of projective schemes, J. Symbolic Comput., 35, 3-19 (2003) · Zbl 1074.14502 [2] Aluffi, P., The Chern-Schwartz-MacPherson class of an embeddable scheme · Zbl 1423.14044 [3] Aluffi, P., The Segre zeta function of an ideal, Adv. Math., 320, 1201-1226 (2017) · Zbl 1376.14028 [4] Aluffi, P.; Faber, E., Chern classes of splayed intersections, Canad. J. Math., 67, 1201-1218 (2015) · Zbl 1328.14011 [5] Barth, W.; Van de Ven, A., A decomposability criterion for algebraic 2-bundles on projective spaces, Invent. Math., 25, 1, 91-106 (1974) · Zbl 0295.14006 [6] Eisenbud, D.; Harris, J., 3264 and All That: A Second Course in Algebraic Geometry (2017), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1341.14001 [7] Eklund, D.; Jost, C.; Peterson, C., A method to compute Segre classes of subschemes of projective space, J. Algebra Appl., 12, 2, Article 1250142 pp. (2013) · Zbl 1274.13044 [8] Fulton, W., Intersection Theory, Ergeb. Math. Grenzgeb., 3. Folge, vol. 2 (1998), Springer: Springer Berlin · Zbl 0885.14002 [9] Grayson, D. R.; Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, Available at [10] Harris, C., Computing segre classes in arbitrary projective varieties, J. Symb. Comput., 82, 26-37 (2017) · Zbl 1357.14070 [11] Hartshorne, R., Algebraic Geometry, Grad. Texts in Math., vol. 52 (1977), Springer-Verlag · Zbl 0367.14001 [12] Hassett, B.; Kovács, S. J., Reflexive pull-backs and base extension, J. Algebraic Geom., 13, 233-247 (2004) · Zbl 1081.14017 [13] Helmer, M., An algorithm to compute the topological Euler characteristic, Chern-Schwartz-MacPherson class and segre class of projective varieties, J. Symb. Comput., 73, 120-138 (2016) · Zbl 1349.14028 [14] Hulek, K., The Horrocks-Mumford bundle, (Hitchin, N.; Newstead, P.; Oxbury, W., Vector Bundles in Algebraic Geometry. Vector Bundles in Algebraic Geometry, London Mathematical Society Lecture Note Series (1995), Cambridge University Press: Cambridge University Press Cambridge), 139-178 · Zbl 0827.14011 [15] Kempf, G. R., A criterion for the splitting of a vector bundle, Forum Math., 2, 5, 477-480 (1990) · Zbl 0715.14004 [16] Kumar, M., Vector bundles on projective spaces, Adv. Alg. Geom., 185-188 (2003) · Zbl 1031.14007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.