January  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. doi: 10.1155/S1073792894000498. Google Scholar

[2]

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

[3]

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

[4]

Paolo Aluffi, Computing characteristic classes of projective schemes,, J. Symbolic Comput., 35 (2003), 3. doi: 10.1016/S0747-7171(02)00089-5. Google Scholar

[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. doi: 10.1016/j.jsc.2010.06.026. Google Scholar

[6]

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

[7]

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

[8]

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

[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/}., (). Google Scholar

[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. doi: 10.4007/annals.2012.176.2.5. Google Scholar

[11]

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

[12]

J. Rodriguez, Combinatorial excess intersection,, \arXiv{1212.2249}, (2012). Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

Paolo Aluffi, Computing characteristic classes of projective schemes,, J. Symbolic Comput., 35 (2003), 3. doi: 10.1016/S0747-7171(02)00089-5. Google Scholar

[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. doi: 10.1016/j.jsc.2010.06.026. Google Scholar

[6]

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

[7]

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

[8]

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

[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/}., (). Google Scholar

[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. doi: 10.4007/annals.2012.176.2.5. Google Scholar

[11]

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

[12]

J. Rodriguez, Combinatorial excess intersection,, \arXiv{1212.2249}, (2012). Google Scholar

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