Segre class computation and practical applications

Harris, Corey; Helmer, Martin

doi:10.1090/mcom/3448

Segre class computation and practical applications
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Math. Comp. 89 (2020), 465-491 Request permission

Abstract:

Let $X \subset Y$ be closed (possibly singular) subschemes of a smooth projective toric variety $T$. We show how to compute the Segre class ${s}(X,Y)$ as a class in the Chow group of $T$. Building on this, we give effective methods to compute intersection products in projective varieties, to determine algebraic multiplicity without working in local rings, and to test pairwise containment of subvarieties of $T$. Our methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used.

References

Paolo Aluffi and Corey Harris, The Euclidean distance degree of smooth complex projective varieties, Algebra Number Theory 12 (2018), no. 8, 2005–2032. MR 3892971, DOI 10.2140/ant.2018.12.2005
Paolo Aluffi, Computing characteristic classes of projective schemes, J. Symbolic Comput. 35 (2003), no. 1, 3–19. MR 1956868, DOI 10.1016/S0747-7171(02)00089-5
P. Aluffi, The Chern-Schwartz-MacPherson class of an embeddable scheme, arXiv preprint arXiv:1805.11116, 2018.
D. J. Bates, J. D. Hauenstein, A. J. Sommese, and C. W. Wampler, Bertini: Software for numerical algebraic geometry, available at bertini.nd.edu with permanent doi: dx.doi.org/10.7274/R0H41PB5.
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322, DOI 10.1090/gsm/124
V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247 (Russian). MR 495499
David Eklund, Christine Jost, and Chris Peterson, A method to compute Segre classes of subschemes of projective space, J. Algebra Appl. 12 (2013), no. 2, 1250142, 15. MR 3005594, DOI 10.1142/S0219498812501423
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2.
Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
Corey Harris, Computing Segre classes in arbitrary projective varieties, J. Symbolic Comput. 82 (2017), 26–37. MR 3608229, DOI 10.1016/j.jsc.2016.09.003
J. D. Hauenstein, M. S. El Din, É. Schost, and T. X. Vu, Solving determinantal systems using homotopy techniques, arXiv preprint arXiv:1802.10409, 2018.
Martin Helmer, Algorithms to compute the topological Euler characteristic, Chern-Schwartz-MacPherson class and Segre class of projective varieties, J. Symbolic Comput. 73 (2016), 120–138. MR 3385954, DOI 10.1016/j.jsc.2015.03.007
Martin Helmer, Computing characteristic classes of subschemes of smooth toric varieties, J. Algebra 476 (2017), 548–582. MR 3608162, DOI 10.1016/j.jalgebra.2016.12.024
Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
G. Lecerf, Kronecker: Polynomial equation system solver, software available at http://www.lix.polytechnique.fr/~lecerf/software/kronecker/index.html; http://www.mathemagix.org/www/geomsolvex/doc/html/index.en.html.
Grégoire Lecerf, Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers, J. Complexity 19 (2003), no. 4, 564–596. MR 1991984, DOI 10.1016/S0885-064X(03)00031-1
Torgunn Karoline Moe and Nikolay Qviller, Segre classes on smooth projective toric varieties, Math. Z. 275 (2013), no. 1-2, 529–548. MR 3101819, DOI 10.1007/s00209-013-1146-9
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
Ragni Piene, Polar classes of singular varieties, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 247–276. MR 510551
P. Samuel, Méthodes d’algèbre abstraite en géométrie algébrique, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 4, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1955 (French). MR 0072531
M. Sayrafi, Computations over local rings in Macaulay2, arXiv preprint arXiv:1710.09830, 2017.
J. Verschelde, Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation, ACM Trans. Math. Software (TOMS) 25 (1999), no. 2, 251–276,.