Families of algebraic and continuous maps to projective space
DOI:
https://doi.org/10.7146/math.scand.a-158293Abstract
We explain how results comparing the homology of spaces of algebraic and continuous maps to projective spaces can be leveraged to compare moduli stacks of families of algebraic and continuous maps.
References
Arapura, D., Algebraic geometry over the complex numbers, Universitext. Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-1809-2
Aumonier, A., The topology of spaces of holomorphic maps to projective space, 2024, arXiv.2402.05500 https://doi.org/10.48550/ARXIV.2402.05500
Ayala, D., Homological stability among moduli spaces of holomorphic curves in complex projective space, 2008, arXiv.0811.2274 https://doi.org/10.48550/ARXIV.0811.2274
Cohen, R. L., and Madsen, I., Surfaces in a background space and the homology of mapping class groups, Algebraic geometry—Seattle 2005. Part 1, 43–76, Proc. Sympos. Pure Math., 80, Part 1, Amer. Math. Soc., Providence, RI, 2009. https://doi.org/10.1090/pspum/080.1/2483932
Earle, C. J., and Eells, J., A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43. https://doi.org/10.4310/jdg/1214428816
Ebert, J., The homotopy type of a topological stack, 2009, arXiv.0901.3295 https://doi.org/10.48550/ARXIV.0901.3295
Ebert, J., and Randal-Williams, O., Semisimplicial spaces, Algebr. Geom. Topol. 19 (2019), no. 4, 2099–2150. https://doi.org/10.2140/agt.2019.19.2099
Grothendieck, A., Techniques de construction en géométrie analytique. I. Description axiomatique de l'espace de Teichmüller et de ses variantes, Séminaire Henri Cartan 13 (1960-1961), no. 1, 1–33.
Grothendieck, A., Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie, Inst. Hautes Études Sci. Publ. Math. No. 32 (1967), 361 pp. http://www.numdam.org/item?id=PMIHES_1967__32__361_0
Grothendieck, A., and Raynaud, M., Revêtements etales et groupe fondamental (SGA1), Exposé XII, Springer Berlin Heidelberg, 1971. https://doi.org/10.1007/bfb0058656
Hinich, V., and Vaintrob, A., Augmented Teichmüller spaces and orbifolds, Selecta Math. (N.S.) 16 (2010), no. 3, 533–629. https://doi.org/10.1007/s00029-010-0027-x
Jansen, M. Ø., Stratified homotopy theory of topological ∞-stacks: A toolbox, J. Pure Appl. Algebra 228 (2024), no. 11, Paper No. 107710, 28 pp. https://doi.org/10.1016/j.jpaa.2024.107710
Kleiman, S. L., The Picard scheme, 2005, arXiv:math/0504020 https://doi.org/10.48550/ARXIV.MATH/0504020
Lazarsfeld, R., Positivity in algebraic geometry I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 48. Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/978-3-642-18808-4
Mostovoy, J., Truncated simplicial resolutions and spaces of rational maps, Q. J. Math. 63 (2012), no. 1, 181–187. https://doi.org/10.1093/qmath/haq031
Segal, G., The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39–72. https://doi.org/10.1007/bf02392088
The Stacks Project Authors, Stacks project, https://stacks.math.columbia.edu/, 2025.
