Refined broccoli invariants
Authors:
Lothar Göttsche and Franziska Schroeter
Journal:
J. Algebraic Geom. 28 (2019), 1-41
DOI:
https://doi.org/10.1090/jag/705
Published electronically:
July 27, 2018
MathSciNet review:
3875360
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We introduce a tropical enumerative invariant depending on a variable $y$ which generalizes the tropical refined Severi degree. We show that this refined broccoli invariant is indeed independent of the point configuration, and that it specializes to a tropical descendant Gromov-Witten invariant for $y=1$ and to the corresponding broccoli invariant for $y=-1$. Furthermore, we define tropical refined descendant Gromov-Witten invariants which equal the corresponding refined broccoli invariants giving a new insight to the nature of broccoli invariants. We discuss various possible generalizations e.g. to refinements of bridge curves and Welschinger curves.
References
- Florian Block and Lothar Göttsche, Refined curve counting with tropical geometry, Compos. Math. 152 (2016), no. 1, 115–151. MR 3453390, DOI https://doi.org/10.1112/S0010437X1500754X
- Florian Block, Andreas Gathmann, and Hannah Markwig, Psi-floor diagrams and a Caporaso-Harris type recursion, Israel J. Math. 191 (2012), no. 1, 405–449. MR 2970875, DOI https://doi.org/10.1007/s11856-011-0216-0
- W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96. MR 1492534, DOI https://doi.org/10.1090/pspum/062.2/1492534
- S. A. Filippini and J. Stoppa, Block-Göttsche invariants from wall-crossing, Compos. Math. 151 (2015), no. 8, 1543–1567. MR 3383167, DOI https://doi.org/10.1112/S0010437X14007994
- Lothar Göttsche and Benjamin Kikwai, Refined node polynomials via long edge graphs, Commun. Number Theory Phys. 10 (2016), no. 2, 193–224. MR 3528834, DOI https://doi.org/10.4310/CNTP.2016.v10.n2.a2
- Andreas Gathmann, Michael Kerber, and Hannah Markwig, Tropical fans and the moduli spaces of tropical curves, Compos. Math. 145 (2009), no. 1, 173–195. MR 2480499, DOI https://doi.org/10.1112/S0010437X08003837
- Andreas Gathmann and Hannah Markwig, The numbers of tropical plane curves through points in general position, J. Reine Angew. Math. 602 (2007), 155–177. MR 2300455, DOI https://doi.org/10.1515/CRELLE.2007.006
- Andreas Gathmann and Hannah Markwig, Kontsevich’s formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008), no. 2, 537–560. MR 2370275, DOI https://doi.org/10.1016/j.aim.2007.08.004
- Andreas Gathmann, Hannah Markwig, and Franziska Schroeter, Broccoli curves and the tropical invariance of Welschinger numbers, Adv. Math. 240 (2013), 520–574. MR 3046318, DOI https://doi.org/10.1016/j.aim.2013.03.004
- L. Göttsche and F. Schroeter, Floor diagrams for refined broccoli curves, in preparation.
- Andreas Gathmann and Franziska Schroeter, Irreducible cycles and points in special position in moduli spaces for tropical curves, Electron. J. Combin. 19 (2012), no. 4, Paper 26, 35. MR 3001663, DOI https://doi.org/10.37236/2862
- Lothar Göttsche and Vivek Shende, Refined curve counting on complex surfaces, Geom. Topol. 18 (2014), no. 4, 2245–2307. MR 3268777, DOI https://doi.org/10.2140/gt.2014.18.2245
- Ilia Itenberg, Viatcheslav Kharlamov, and Eugenii Shustin, A Caporaso-Harris type formula for Welschinger invariants of real toric del Pezzo surfaces, Comment. Math. Helv. 84 (2009), no. 1, 87–126. MR 2466076, DOI https://doi.org/10.4171/CMH/153
- Ilia Itenberg and Grigory Mikhalkin, On Block-Göttsche multiplicities for planar tropical curves, Int. Math. Res. Not. IMRN 23 (2013), 5289–5320. MR 3142257, DOI https://doi.org/10.1093/imrn/rns207
- M. Kontsevich and Yu. Manin, Relations between the correlators of the topological sigma-model coupled to gravity, Comm. Math. Phys. 196 (1998), no. 2, 385–398. MR 1645019, DOI https://doi.org/10.1007/s002200050426
- Grigory Mikhalkin, Quantum indices and refined enumeration of real plane curves, Acta Math. 219 (2017), no. 1, 135–180. MR 3765660, DOI https://doi.org/10.4310/ACTA.2017.v219.n1.a5
- Grigory Mikhalkin, Enumerative tropical algebraic geometry in $\Bbb R^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377. MR 2137980, DOI https://doi.org/10.1090/S0894-0347-05-00477-7
- Grigory Mikhalkin, Tropical geometry and its applications, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 827–852. MR 2275625
- Hannah Markwig and Johannes Rau, Tropical descendant Gromov-Witten invariants, Manuscripta Math. 129 (2009), no. 3, 293–335. MR 2515486, DOI https://doi.org/10.1007/s00229-009-0256-5
- J. Nicaise, S. Payne, and F. Schroeter, Tropical refined curve counting via motivic integration, arXiv:1603.08424, 2018.
- Eugenii Shustin, A tropical calculation of the Welschinger invariants of real toric del Pezzo surfaces, J. Algebraic Geom. 15 (2006), no. 2, 285–322. MR 2199066, DOI https://doi.org/10.1090/S1056-3911-06-00434-6
- Jean-Yves Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry, C. R. Math. Acad. Sci. Paris 336 (2003), no. 4, 341–344 (English, with English and French summaries). MR 1976315, DOI https://doi.org/10.1016/S1631-073X%2803%2900059-1
- Jean-Yves Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195–234. MR 2198329, DOI https://doi.org/10.1007/s00222-005-0445-0
References
- Florian Block and Lothar Göttsche, Refined curve counting with tropical geometry, Compos. Math. 152 (2016), no. 1, 115–151. MR 3453390, DOI https://doi.org/10.1112/S0010437X1500754X
- Florian Block, Andreas Gathmann, and Hannah Markwig, Psi-floor diagrams and a Caporaso–Harris type recursion, Israel J. Math. 191 (2012), no. 1, 405–449. MR 2970875, DOI https://doi.org/10.1007/s11856-011-0216-0
- W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96. MR 1492534, DOI https://doi.org/10.1090/pspum/062.2/1492534
- S. A. Filippini and J. Stoppa, Block-Göttsche invariants from wall-crossing, Compos. Math. 151 (2015), no. 8, 1543–1567. MR 3383167, DOI https://doi.org/10.1112/S0010437X14007994
- Lothar Göttsche and Benjamin Kikwai, Refined node polynomials via long edge graphs, Commun. Number Theory Phys. 10 (2016), no. 2, 193–224. MR 3528834, DOI https://doi.org/10.4310/CNTP.2016.v10.n2.a2
- Andreas Gathmann, Michael Kerber, and Hannah Markwig, Tropical fans and the moduli spaces of tropical curves, Compos. Math. 145 (2009), no. 1, 173–195. MR 2480499, DOI https://doi.org/10.1112/S0010437X08003837
- Andreas Gathmann and Hannah Markwig, The numbers of tropical plane curves through points in general position, J. Reine Angew. Math. 602 (2007), 155–177. MR 2300455, DOI https://doi.org/10.1515/CRELLE.2007.006
- Andreas Gathmann and Hannah Markwig, Kontsevich’s formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008), no. 2, 537–560. MR 2370275, DOI https://doi.org/10.1016/j.aim.2007.08.004
- Andreas Gathmann, Hannah Markwig, and Franziska Schroeter, Broccoli curves and the tropical invariance of Welschinger numbers, Adv. Math. 240 (2013), 520–574. MR 3046318, DOI https://doi.org/10.1016/j.aim.2013.03.004
- L. Göttsche and F. Schroeter, Floor diagrams for refined broccoli curves, in preparation.
- Andreas Gathmann and Franziska Schroeter, Irreducible cycles and points in special position in moduli spaces for tropical curves, Electron. J. Combin. 19 (2012), no. 4, Paper 26, 35 pp. MR 3001663
- Lothar Göttsche and Vivek Shende, Refined curve counting on complex surfaces, Geom. Topol. 18 (2014), no. 4, 2245–2307. MR 3268777, DOI https://doi.org/10.2140/gt.2014.18.2245
- Ilia Itenberg, Viatcheslav Kharlamov, and Eugenii Shustin, A Caporaso-Harris type formula for Welschinger invariants of real toric del Pezzo surfaces, Comment. Math. Helv. 84 (2009), no. 1, 87–126. MR 2466076, DOI https://doi.org/10.4171/CMH/153
- Ilia Itenberg and Grigory Mikhalkin, On Block-Göttsche multiplicities for planar tropical curves, Int. Math. Res. Not. IMRN 23 (2013), 5289–5320. MR 3142257
- M. Kontsevich and Yu. Manin, Relations between the correlators of the topological sigma-model coupled to gravity, Comm. Math. Phys. 196 (1998), no. 2, 385–398. MR 1645019, DOI https://doi.org/10.1007/s002200050426
- Grigory Mikhalkin, Quantum indices and refined enumeration of real plane curves, Acta Math. 219 (2017), no. 1, 135–180. MR 3765660, DOI https://doi.org/10.4310/ACTA.2017.v219.n1.a5
- Grigory Mikhalkin, Enumerative tropical algebraic geometry in $\mathbb {R}^2$, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377. MR 2137980, DOI https://doi.org/10.1090/S0894-0347-05-00477-7
- Grigory Mikhalkin, Tropical geometry and its applications, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 827–852. MR 2275625
- Hannah Markwig and Johannes Rau, Tropical descendant Gromov–Witten invariants, Manuscripta Math. 129 (2009), no. 3, 293–335. MR 2515486, DOI https://doi.org/10.1007/s00229-009-0256-5
- J. Nicaise, S. Payne, and F. Schroeter, Tropical refined curve counting via motivic integration, arXiv:1603.08424, 2018.
- Eugenii Shustin, A tropical calculation of the Welschinger invariants of real toric del Pezzo surfaces, J. Algebraic Geom. 15 (2006), no. 2, 285–322. MR 2199066, DOI https://doi.org/10.1090/S1056-3911-06-00434-6
- Jean-Yves Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry, C. R. Math. Acad. Sci. Paris 336 (2003), no. 4, 341–344 (English, with English and French summaries). MR 1976315, DOI https://doi.org/10.1016/S1631-073X%2803%2900059-1
- Jean-Yves Welschinger, Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry, Invent. Math. 162 (2005), no. 1, 195–234. MR 2198329, DOI https://doi.org/10.1007/s00222-005-0445-0
Additional Information
Lothar Göttsche
Affiliation:
International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
MR Author ID:
288886
Email:
gottsche@ictp.it
Franziska Schroeter
Affiliation:
Fachbereich Mathematik (AD), Universitát Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany
MR Author ID:
1003319
Email:
franziska.schroeter@uni-hamburg.de
Received by editor(s):
August 4, 2016
Received by editor(s) in revised form:
April 14, 2017, and April 27, 2017
Published electronically:
July 27, 2018
Additional Notes:
The second author was partially supported by GIF grant No. 1174-197.6/2011, the Minkowski-Minerva Center for Geometry at the Tel Aviv University, by grant No. 178/13 from the Israel Science Foundation, and by the RTG 1670 “Mathematics Inspired by String Theory and Quantum Field Theory” funded by the German Research Foundation (DFG)
Article copyright:
© Copyright 2018
University Press, Inc.
