Enumeration of algebraic and tropical singular hypersurfaces

Sinichkin, Uriel

doi:10.1090/tran/8753

Enumeration of algebraic and tropical singular hypersurfaces
HTML articles powered by AMS MathViewer

by Uriel Sinichkin PDF

Trans. Amer. Math. Soc. 375 (2022), 8529-8580

Abstract:

We develop a version of Mikhalkin’s lattice path algorithm for projective hypersurfaces of arbitrary degree and dimension, which enumerates singular tropical hypersurfaces passing through appropriate configuration of points. By proving a correspondence theorem combined with the lattice path algorithm, we construct a $\delta$ dimensional linear space of degree $d$ real hypersurfaces containing $\frac {1}{\delta !}(\gamma _nd^n)^{\delta }+O(d^{n\delta -1})$ hypersurfaces with $\delta$ real nodes, where $\gamma _n$ are positive and given by a recursive formula. This is asymptotically comparable to the number $\frac {1}{\delta !} \left ( (n+1)(d-1)^n \right )^{\delta }+O\left (d^{n(\delta -1)} \right )$ of complex hypersurfaces having $\delta$ nodes in a $\delta$ dimensional linear space. In the case $\delta =1$ we give a slightly better leading term.

References

Alicia Dickenstein and Luis F. Tabera, Singular tropical hypersurfaces, Discrete Comput. Geom. 47 (2012), no. 2, 430–453. MR 2872547, DOI 10.1007/s00454-011-9364-6
I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants and multidimensional determinants, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008. Reprint of the 1994 edition. MR 2394437
Branko Grünbaum, Convex polytopes, 2nd ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler. MR 1976856, DOI 10.1007/978-1-4613-0019-9
Ó. Iglesias Valiño and F. Santos, The complete classification of empty lattice 4-simplices, Electron. Notes Discrete Math. 68 (2018), 155–160.
Ilia Itenberg, Grigory Mikhalkin, and Eugenii Shustin, Tropical algebraic geometry, 2nd ed., Oberwolfach Seminars, vol. 35, Birkhäuser Verlag, Basel, 2009. MR 2508011, DOI 10.1007/978-3-0346-0048-4
Christian U. Jensen and Helmut Lenzing, Model-theoretic algebra with particular emphasis on fields, rings, modules, Algebra, Logic and Applications, vol. 2, Gordon and Breach Science Publishers, New York, 1989. MR 1057608
D. Kerner, Personal communication, 2017.
Diane Maclagan and Bernd Sturmfels, Introduction to tropical geometry, Graduate Studies in Mathematics, vol. 161, American Mathematical Society, Providence, RI, 2015. MR 3287221, DOI 10.1090/gsm/161
H. Markwig, T. Markwig, and E. Shustin, Tropical curves with a singularity in a fixed point, Manuscripta Math. 137 (2012), no. 3, 383–418.
Hannah Markwig, Thomas Markwig, and Eugenii Shustin, Tropical curves with a singularity in a fixed point, Manuscripta Math. 137 (2012), no. 3-4, 383–418. MR 2875284, DOI 10.1007/s00229-011-0471-8
Hannah Markwig, Thomas Markwig, and Eugenii Shustin, Tropical surface singularities, Discrete Comput. Geom. 48 (2012), no. 4, 879–914. MR 3000569, DOI 10.1007/s00454-012-9453-1
Hannah Markwig, Thomas Markwig, and Eugenii Shustin, Enumeration of complex and real surfaces via tropical geometry, Adv. Geom. 18 (2018), no. 1, 69–100. MR 3750255, DOI 10.1515/advgeom-2017-0024
Grigory Mikhalkin, Counting curves via lattice paths in polygons, C. R. Math. Acad. Sci. Paris 336 (2003), no. 8, 629–634 (English, with English and French summaries). MR 1988122, DOI 10.1016/S1631-073X(03)00104-3
Grigory Mikhalkin, Enumerative tropical algebraic geometry in $\Bbb R^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377. MR 2137980, DOI 10.1090/S0894-0347-05-00477-7
E. Shustin, A tropical approach to enumerative geometry, Algebra i Analiz 17 (2005), no. 2, 170–214; English transl., St. Petersburg Math. J. 17 (2006), no. 2, 343–375. MR 2159589, DOI 10.1090/S1061-0022-06-00908-3
O. Viro, Patchworking Real Algebraic Varieties, Appendix to G.-M. Greuel et al. Singular Algebraic Curves, Springer, Switzerland, 2018.
G. M. Ziegler, Lectures On Polytopes, Grad. Texts in Math., Springer, 2007.