Segre classes of monomial schemes

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

Segre classes of monomial schemes

Paolo Aluffi ^1,

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

Received February 2013 Revised May 2013 Published May 2013

Abstract
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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.

Mathematics Subject Classification: 14C1.

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).

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