2013, 20: 55-70. doi: 10.3934/era.2013.20.55

Segre classes of monomial schemes

1. 

Mathematics Department, Florida State University, Tallahassee FL 32306, United States

Received  February 2013 Revised  May 2013 Published  May 2013

We propose an explicit formula for the Segre classes of monomial subschemes of nonsingular varieties, such as schemes defined by monomial ideals in projective space. The Segre class is expressed as a formal integral on a region bounded by the corresponding Newton polyhedron. We prove this formula for monomial ideals in two variables and verify it for some families of examples in any number of variables.
Citation: Paolo Aluffi. Segre classes of monomial schemes. Electronic Research Announcements, 2013, 20: 55-70. doi: 10.3934/era.2013.20.55
References:
[1]

Paolo Aluffi, MacPherson's and Fulton's Chern classes of hypersurfaces, Internat. Math. Res. Notices, 1994 (1994), 455-465. doi: 10.1155/S1073792894000498.

[2]

Paolo Aluffi, Singular schemes of hypersurfaces, Duke Math. J., 80 (1995), 325-351. doi: 10.1215/S0012-7094-95-08014-4.

[3]

Paolo Aluffi, Chern classes for singular hypersurfaces, Trans. Amer. Math. Soc., 351 (1999), 3989-4026. doi: 10.1090/S0002-9947-99-02256-4.

[4]

Paolo Aluffi, Computing characteristic classes of projective schemes, J. Symbolic Comput., 35 (2003), 3-19. Code available from: http://www.math.fsu.edu/~aluffi/CSM/CSM.html. doi: 10.1016/S0747-7171(02)00089-5.

[5]

Sandra Di Rocco, David Eklund, Chris Peterson and Andrew J. Sommese, Chern numbers of smooth varieties via homotopy continuation and intersection theory, J. Symbolic Comput., 46 (2011), 23-33. doi: 10.1016/j.jsc.2010.06.026.

[6]

David Eklund, Christine Jost and Chris Peterson, A method to compute Segre classes of subschemes of projective space, J. Algebra Appl., 12 (2013), 1250142, 15 pp. doi: 10.1142/S0219498812501423.

[7]

William Fulton, "Intersection Theory," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 2, Springer-Verlag, Berlin, 1984.

[8]

Russell A. Goward, Jr., A simple algorithm for principalization of monomial ideals, Trans. Amer. Math. Soc., 357 (2005), 4805-4812 (electronic). doi: 10.1090/S0002-9947-05-03866-3.

[9]

Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry., Available from: \url{http://www.math.uiuc.edu/Macaulay2/}., (). 

[10]

Kiumars Kaveh and A. G. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2), 176 (2012), 925-978. doi: 10.4007/annals.2012.176.2.5.

[11]

Robert Lazarsfeld and Mircea Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 783-835.

[12]

J. Rodriguez, Combinatorial excess intersection, arXiv:1212.2249, (2012).

show all references

References:
[1]

Paolo Aluffi, MacPherson's and Fulton's Chern classes of hypersurfaces, Internat. Math. Res. Notices, 1994 (1994), 455-465. doi: 10.1155/S1073792894000498.

[2]

Paolo Aluffi, Singular schemes of hypersurfaces, Duke Math. J., 80 (1995), 325-351. doi: 10.1215/S0012-7094-95-08014-4.

[3]

Paolo Aluffi, Chern classes for singular hypersurfaces, Trans. Amer. Math. Soc., 351 (1999), 3989-4026. doi: 10.1090/S0002-9947-99-02256-4.

[4]

Paolo Aluffi, Computing characteristic classes of projective schemes, J. Symbolic Comput., 35 (2003), 3-19. Code available from: http://www.math.fsu.edu/~aluffi/CSM/CSM.html. doi: 10.1016/S0747-7171(02)00089-5.

[5]

Sandra Di Rocco, David Eklund, Chris Peterson and Andrew J. Sommese, Chern numbers of smooth varieties via homotopy continuation and intersection theory, J. Symbolic Comput., 46 (2011), 23-33. doi: 10.1016/j.jsc.2010.06.026.

[6]

David Eklund, Christine Jost and Chris Peterson, A method to compute Segre classes of subschemes of projective space, J. Algebra Appl., 12 (2013), 1250142, 15 pp. doi: 10.1142/S0219498812501423.

[7]

William Fulton, "Intersection Theory," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 2, Springer-Verlag, Berlin, 1984.

[8]

Russell A. Goward, Jr., A simple algorithm for principalization of monomial ideals, Trans. Amer. Math. Soc., 357 (2005), 4805-4812 (electronic). doi: 10.1090/S0002-9947-05-03866-3.

[9]

Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry., Available from: \url{http://www.math.uiuc.edu/Macaulay2/}., (). 

[10]

Kiumars Kaveh and A. G. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2), 176 (2012), 925-978. doi: 10.4007/annals.2012.176.2.5.

[11]

Robert Lazarsfeld and Mircea Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 783-835.

[12]

J. Rodriguez, Combinatorial excess intersection, arXiv:1212.2249, (2012).

[1]

Armengol Gasull, Jaume Giné, Joan Torregrosa. Center problem for systems with two monomial nonlinearities. Communications on Pure and Applied Analysis, 2016, 15 (2) : 577-598. doi: 10.3934/cpaa.2016.15.577

[2]

Francis N. Castro, Carlos Corrada-Bravo, Natalia Pacheco-Tallaj, Ivelisse Rubio. Explicit formulas for monomial involutions over finite fields. Advances in Mathematics of Communications, 2017, 11 (2) : 301-306. doi: 10.3934/amc.2017022

[3]

Dorothy Bollman, Omar Colón-Reyes. Determining steady state behaviour of discrete monomial dynamical systems. Advances in Mathematics of Communications, 2017, 11 (2) : 283-287. doi: 10.3934/amc.2017019

[4]

Kathy Horadam, Russell East. Partitioning CCZ classes into EA classes. Advances in Mathematics of Communications, 2012, 6 (1) : 95-106. doi: 10.3934/amc.2012.6.95

[5]

Xiao Wen. Structurally stable homoclinic classes. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1693-1707. doi: 10.3934/dcds.2016.36.1693

[6]

Jana Rodriguez Hertz, Carlos H. Vásquez. Structure of accessibility classes. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4653-4664. doi: 10.3934/dcds.2020196

[7]

Christian Bonatti, Shaobo Gan, Dawei Yang. On the hyperbolicity of homoclinic classes. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143

[8]

Marx Chhay, Aziz Hamdouni. On the accuracy of invariant numerical schemes. Communications on Pure and Applied Analysis, 2011, 10 (2) : 761-783. doi: 10.3934/cpaa.2011.10.761

[9]

Benjamin Seibold, Rodolfo R. Rosales, Jean-Christophe Nave. Jet schemes for advection problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1229-1259. doi: 10.3934/dcdsb.2012.17.1229

[10]

Laurenţiu Maxim, Jörg Schürmann. Characteristic classes of singular toric varieties. Electronic Research Announcements, 2013, 20: 109-120. doi: 10.3934/era.2013.20.109

[11]

Pieter Moree. On the distribution of the order over residue classes. Electronic Research Announcements, 2006, 12: 121-128.

[12]

Yiwei Liu, Zihui Liu. On some classes of codes with a few weights. Advances in Mathematics of Communications, 2018, 12 (2) : 415-428. doi: 10.3934/amc.2018025

[13]

Yvo Desmedt, Niels Duif, Henk van Tilborg, Huaxiong Wang. Bounds and constructions for key distribution schemes. Advances in Mathematics of Communications, 2009, 3 (3) : 273-293. doi: 10.3934/amc.2009.3.273

[14]

Yakov Pesin, Samuel Senti. Equilibrium measures for maps with inducing schemes. Journal of Modern Dynamics, 2008, 2 (3) : 397-430. doi: 10.3934/jmd.2008.2.397

[15]

Claire david@lmm.jussieu.fr David, Pierre Sagaut. Theoretical optimization of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 286-293. doi: 10.3934/proc.2007.2007.286

[16]

Eunkyoung Ko, Eun Kyoung Lee, R. Shivaji. Multiplicity results for classes of singular problems on an exterior domain. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5153-5166. doi: 10.3934/dcds.2013.33.5153

[17]

S. Bautista, C. Morales, M. J. Pacifico. On the intersection of homoclinic classes on singular-hyperbolic sets. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 761-775. doi: 10.3934/dcds.2007.19.761

[18]

Dung Tien Nguyen, Son Luu Nguyen, Nguyen Huu Du. On mean field systems with multi-classes. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 683-707. doi: 10.3934/dcds.2020057

[19]

Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911

[20]

Eun Kyoung Lee, R. Shivaji, Jinglong Ye. Classes of singular $pq-$Laplacian semipositone systems. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1123-1132. doi: 10.3934/dcds.2010.27.1123

2020 Impact Factor: 0.929

Metrics

  • PDF downloads (75)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]