Tropical floor plans and enumeration of complex and real multi-nodal surfaces

Authors:
Hannah Markwig, Thomas Markwig, Kris Shaw and Eugenii Shustin

Journal:
J. Algebraic Geom. **31** (2022), 261-301

DOI:
https://doi.org/10.1090/jag/774

Published electronically:
November 2, 2021

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Abstract |
References |
Additional Information

Abstract:

The family of complex projective surfaces in $\mathbb {P}^3$ of degree $d$ having precisely $\delta$ nodes as their only singularities has codimension $\delta$ in the linear system $|{\mathcal O}_{\mathbb {P}^3}(d)|$ for sufficiently large $d$ and is of degree $N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d)=(4(d-1)^3)^\delta /\delta !+O(d^{3\delta -3})$. In particular, $N_{\delta ,\mathbb {C}}^{\mathbb {P}^3}(d)$ is polynomial in $d$.

By means of tropical geometry, we explicitly describe $(4d^3)^\delta /\delta !+O(d^{3\delta -1})$ surfaces passing through a suitable generic configuration of $n=\binom {d+3}{3}-\delta -1$ points in $\mathbb {P}^3$. These surfaces are close to tropical limits which we characterize combinatorially, introducing the concept of floor plans for multinodal tropical surfaces. The concept of floor plans is similar to the well-known floor diagrams (a combinatorial tool for tropical curve counts): with it, we keep the combinatorial essentials of a multinodal tropical surface $S$ which are sufficient to reconstruct $S$.

In the real case, we estimate the range for possible numbers of real multi-nodal surfaces satisfying point conditions. We show that, for a special configuration $\boldsymbol {w}$ of real points, the number $N_{\delta ,\mathbb {R}}^{\mathbb {P}^3}(d,\boldsymbol {w})$ of real surfaces of degree $d$ having $\delta$ real nodes and passing through $\boldsymbol {w}$ is bounded from below by $\left (\frac {3}{2}d^3\right )^\delta /\delta ! +O(d^{3\delta -1})$.

We prove analogous statements for counts of multinodal surfaces in $\mathbb {P}^1\times \mathbb {P}^2$ and $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^1$.

References
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*Énumération des courbes réelles: étude de la maximalité*, In preparation.
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*A tropical count of binodal cubic surfaces*, Le Matematiche **75** (2020), no. 2, 627–649.
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*Labeled floor diagrams for plane curves*, J. Eur. Math. Soc. (JEMS) **12** (2010), no. 6, 1453–1496. MR **2734349**, DOI 10.4171/JEMS/238
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*Tropical algebraic geometry*, 2nd ed., Oberwolfach Seminars, vol. 35, Birkhäuser Verlag, Basel, 2009. MR **2508011**, DOI 10.1007/978-3-0346-0048-4
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*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
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*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
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*Versal deformation of algebraic hypersurfaces with isolated singularities*, Math. Ann. **313** (1999), no. 2, 297–314. MR **1679787**, DOI 10.1007/s002080050262
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*Enumeration of algebraic and tropical singular hypersurfaces*, Master Thesis, Tel Aviv University, 2020.
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References
- Benoit Bertrand,
*Énumération des courbes réelles: étude de la maximalité*, In preparation.
- Benoît Bertrand, Erwan Brugallé, and Lucía López de Medrano,
*Planar tropical cubic curves of any genus, and higher dimensional generalisations*, Enseign. Math. **64** (2018), no. 3-4, 415–457. MR **3987152**, DOI 10.4171/LEM/64-3/4-10
- Madeline Brandt and Alheydis Geiger,
*A tropical count of binodal cubic surfaces*, Le Matematiche **75** (2020), no. 2, 627–649.
- Erwan Brugallé and Grigory Mikhalkin,
*Floor decompositions of tropical curves: the planar case*, Proceedings of Gökova Geometry-Topology Conference 2008, Gökova Geometry/Topology Conference (GGT), Gökova, 2009, pp. 64–90. MR **2500574**
- A. A. du Plessis and C. T. C. Wall,
*Singular hypersurfaces, versality, and Gorenstein algebras*, J. Algebraic Geom. **9** (2000), no. 2, 309–322. MR **1735775**
- Sergey Fomin and Grigory Mikhalkin,
*Labeled floor diagrams for plane curves*, J. Eur. Math. Soc. (JEMS) **12** (2010), no. 6, 1453–1496. MR **2734349**, DOI 10.4171/JEMS/238
- 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
- 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,
*Enumerative tropical algebraic geometry in $\mathbb {R}^2$*, J. Amer. Math. Soc. **18** (2005), no. 2, 313–377. MR **2137980**, DOI 10.1090/S0894-0347-05-00477-7
- Eugenii Shustin and Ilya Tyomkin,
*Versal deformation of algebraic hypersurfaces with isolated singularities*, Math. Ann. **313** (1999), no. 2, 297–314. MR **1679787**, DOI 10.1007/s002080050262
- Uriel Sinichkin,
*Enumeration of algebraic and tropical singular hypersurfaces*, Master Thesis, Tel Aviv University, 2020.
- O. Ya. Viro,
*Some integral calculus based on Euler characteristic*, Topology and geometry—Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer, Berlin, 1988, pp. 127–138. MR **970076**, DOI 10.1007/BFb0082775

Additional Information

**Hannah Markwig**

Affiliation:
Eberhard Karls Universität Tübingen, Fachbereich Mathematik, Auf der Morgenstelle 10, 72076 Tübingen, Germany

MR Author ID:
741165

Email:
hannah@math.uni-tuebingen.de

**Thomas Markwig**

Affiliation:
Eberhard Karls Universität Tübingen, Fachbereich Mathematik, Auf der Morgenstelle 10, 72076 Tübingen, Germany

MR Author ID:
689521

Email:
keilen@math.uni-tuebingen.de

**Kris Shaw**

Affiliation:
Department of Mathematics, University of Oslo, Postboks 1053 Blindern, 0316 Oslo, Norway

Email:
krisshaw@math.uio.no

**Eugenii Shustin**

Affiliation:
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 6997801, Israel

MR Author ID:
193452

Email:
shustin@tauex.tau.ac.il

Received by editor(s):
November 1, 2019

Received by editor(s) in revised form:
August 20, 2020

Published electronically:
November 2, 2021

Additional Notes:
The first and second authors were partially supported by the DFG-collaborative research center TRR 195 (INST 248/235-1). The research of the third author was supported by the Bergen Research Foundation project “Algebraic and topological cycles in complex and tropical geometry”. The fourth author was supported by the Israeli Science Foundation grants no. 176/15 and 501/18 and by the Bauer-Neuman Chair in Real and Complex Geometry.

Article copyright:
© Copyright 2021
University Press, Inc.