Algebra & Number Theory

The Euclidean distance degree of smooth complex projective varieties

Paolo Aluffi and Corey Harris

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern–Schwartz–MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of X with the Euler characteristic of an open subset of X.

Article information

Source
Algebra Number Theory, Volume 12, Number 8 (2018), 2005-2032.

Dates
Received: 3 November 2017
Revised: 3 May 2018
Accepted: 19 June 2018
First available in Project Euclid: 21 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1545361468

Digital Object Identifier
doi:10.2140/ant.2018.12.2005

Mathematical Reviews number (MathSciNet)
MR3892971

Zentralblatt MATH identifier
06999508

Subjects
Primary: 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
Secondary: 14N10: Enumerative problems (combinatorial problems) 57R20: Characteristic classes and numbers

Keywords
algebraic optimization intersection theory characteristic classes Chern–Schwartz–MacPherson classes

Citation

Aluffi, Paolo; Harris, Corey. The Euclidean distance degree of smooth complex projective varieties. Algebra Number Theory 12 (2018), no. 8, 2005--2032. doi:10.2140/ant.2018.12.2005. https://projecteuclid.org/euclid.ant/1545361468


Export citation

References

  • P. Aluffi, “MacPherson's and Fulton's Chern classes of hypersurfaces”, Internat. Math. Res. Notices 11 (1994), 455–465.
  • P. Aluffi, “Singular schemes of hypersurfaces”, Duke Math. J. 80:2 (1995), 325–351.
  • P. Aluffi, “Chern classes for singular hypersurfaces”, Trans. Amer. Math. Soc. 351:10 (1999), 3989–4026.
  • P. Aluffi, “Differential forms with logarithmic poles and Chern–Schwartz–MacPherson classes of singular varieties”, C. R. Acad. Sci. Paris Sér. I Math. 329:7 (1999), 619–624.
  • P. Aluffi, “Weighted Chern–Mather classes and Milnor classes of hypersurfaces”, pp. 1–20 in Singularities - Sapporo $1998$, edited by J.-P. Brasselet and T. Suwa, Adv. Stud. Pure Math. 29, Kinokuniya, Tokyo, 2000.
  • P. Aluffi, “Computing characteristic classes of projective schemes”, J. Symbolic Comput. 35:1 (2003), 3–19.
  • P. Aluffi, “Euler characteristics of general linear sections and polynomial Chern classes”, Rend. Circ. Mat. Palermo $(2)$ 62:1 (2013), 3–26.
  • P. Aluffi, “Tensored Segre classes”, J. Pure Appl. Algebra 221:6 (2017), 1366–1382.
  • P. Aluffi, “Projective duality and a Chern–Mather involution”, Trans. Amer. Math. Soc. 370:3 (2018), 1803–1822.
  • J. Draisma, E. Horobe\commaaccentt, G. Ottaviani, B. Sturmfels, and R. R. Thomas, “The Euclidean distance degree of an algebraic variety”, Found. Comput. Math. 16:1 (2016), 99–149.
  • S. Friedland and G. Ottaviani, “The number of singular vector tuples and uniqueness of best rank-one approximation of tensors”, Found. Comput. Math. 14:6 (2014), 1209–1242.
  • W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 2, Springer, 1984.
  • M. Goresky and W. Pardon, “Chern classes of automorphic vector bundles”, Invent. Math. 147:3 (2002), 561–612.
  • C. Harris, “Computing Segre classes in arbitrary projective varieties”, J. Symbolic Comput. 82 (2017), 26–37.
  • M. 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.
  • C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series 336, Cambridge University Press, 2006.
  • C. Jost, “An algorithm for computing the topological Euler characteristic of complex projective varieties”, 2013.
  • C. Jost, “Computing characteristic classes and the topological Euler characteristic of complex projective schemes”, J. Softw. Algebra Geom. 7 (2015), 31–39.
  • H. Lee, “The Euclidean distance degree of Fermat hypersurfaces”, J. Symbolic Comput. 80:2 (2017), 502–510.
  • X. Liao, “Chern classes of logarithmic vector fields for locally-homogenous free divisors”, 2012.
  • X. Liao, “Chern classes of logarithmic derivations for free divisors with Jacobian ideal of linear type”, J. Math. Soc. Japan 70:3 (2018), 975–988.
  • D. R. Grayson and M. E. Stillman, “Macaulay2, a software system for research in algebraic geometry”, http://www.math.uiuc.edu/Macaulay2/.
  • R. D. MacPherson, “Chern classes for singular algebraic varieties”, Ann. of Math. $(2)$ 100 (1974), 423–432.
  • M. A. Marco-Buzunáriz, “A polynomial generalization of the Euler characteristic for algebraic sets”, J. Singul. 4 (2012), 114–130.
  • A. Parusiński and P. Pragacz, “Characteristic classes of hypersurfaces and characteristic cycles”, J. Algebraic Geom. 10:1 (2001), 63–79.
  • The Sage developers, “Sagemath, the Sage mathematics software system”, http://www.sagemath.org.
  • J. Schürmann, “Chern classes and transversality for singular spaces”, pp. 207–231 in Singularities in geometry, topology, foliations and dynamics, edited by J. L. Cisneros-Molina et al., Springer, Cham, Switzerland, 2017.
  • M.-H. Schwartz, “Classes caractéristiques définies par une stratification d'une variété analytique complexe I”, C. R. Acad. Sci. Paris 260 (1965), 3262–3264.
  • M.-H. Schwartz, “Classes caractéristiques définies par une stratification d'une variété analytique complexe II”, C. R. Acad. Sci. Paris 260 (1965), 3535–3537.