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EIGENPERIODS AND THE MODULI OF POINTS IN THE LINE
Published online by Cambridge University Press: 24 February 2025
Abstract
We study the period map of configurations of n points on the projective line constructed via a cyclic cover branching along these points. By considering the decomposition of its Hodge structure into eigenspaces, we establish the codimension of the locus where the eigenperiod map is still pure. Furthermore, we show that the period map extends to the divisors of a specific moduli space of weighted stable rational curves, and that this extension satisfies a local Torelli map along its fibers.
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References
Ash, A., Mumford, D., Rapoport, M. and Tai, Y. S., Smooth compactifications of locally symmetric varieties, 2nd ed., Cambridge University Press, Cambridge, 2010.CrossRefGoogle Scholar
Carlson, J. A., Extensions of mixed Hodge structures. Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979. Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980, pp. 107–127.Google Scholar
Castor, B., Deng, H., Kerr, M. and Pearlstein, G.,
Remarks on eigenspectra of isolated singularities
, Pac. J. Math. 327 (2023), no. 1, 29–54.CrossRefGoogle Scholar
Chen, L, Gibney, A and Krashen, D,
Pointed trees of projective spaces
, J. Algebr. Geom. 18 (2009), no. 3, 477–509.CrossRefGoogle Scholar
Casalaina-Martin, S., Grushevsky, S. and Hulek, K., The birational geometry of moduli of cubic surfaces and cubic surfaces with a line, preprint, arXiv preprint arXiv:2312.15369, 2023.Google Scholar
Dolgachev, I. V. and Kondō, S., “Moduli of K3 surfaces and complex ball quotients” in R.-P. Holzapfel, A. Muhammed Uludağ, M. Yoshida (eds.), Arithmetic and geometry around hypergeometric functions, Springer, Berlin, 2007, 43–100.CrossRefGoogle Scholar
Deligne, P. and Mostow, G. D.,
Monodromy of hypergeometric functions and non-lattice integral monodromy
, Publ. Math. l’Inst. Hautes Études Sci. 63 (1986), 5–89.CrossRefGoogle Scholar
Doran, B. R., Hurwitz spaces and moduli spaces as ball quotients via pull-back, preprint, arXiv preprint math/0404363, 2004.Google Scholar
Fedorchuk, M., Cyclic covering morphisms on
${\overline{M}}_{0,n}$
, preprint, arXiv preprint arXiv:1105.0655, 17, 2011.Google Scholar
${\overline{M}}_{0,n}$
, preprint, arXiv preprint arXiv:1105.0655, 17, 2011.Google ScholarGallardo, P., Kerr, M. and Schaffler, L.,
Geometric interpretation of toroidal compactifications of moduli of points in the line and cubic surfaces
, Adv. Math. 381 (2021), 107632.CrossRefGoogle Scholar
Gallardo, P. and Routis, E.,
Wonderful compactifications of the moduli space of points in affine and projective space
, Eur. J. Math. 3 (2017), 520–564.CrossRefGoogle Scholar
Griffiths, P. A.,
Periods of integrals on algebraic manifolds, iii. some global differential-geometric properties of the period mapping
, Publ. Math. Inst. des Hautes Études Sci. 38 (1970), 125–180.CrossRefGoogle Scholar
Hassett, B.,
Moduli spaces of weighted pointed stable curves
, Adv. Math. 173 (2003), no. 2, 316–352.CrossRefGoogle Scholar
Hulek, K., Kondo, S. and Maeda, Y., Compactifications of the Eisenstein ancestral Deligne-Mostow variety, preprint, arXiv preprint arXiv:2403.18345, 2024.Google Scholar
Hulek, K. and Maeda, Y., Revisiting the moduli space of 8 points on
${\mathbb{P}}^1$
, preprint, arXiv preprint arXiv:2211.00052, 2022.Google Scholar
${\mathbb{P}}^1$
, preprint, arXiv preprint arXiv:2211.00052, 2022.Google ScholarKirwan, F. C.,
Partial desingularisations of quotients of nonsingular varieties and their betti numbers
, Ann. Math. 122 (1985), no. 1, 41–q85.CrossRefGoogle Scholar
Kerr, M. and Laza, R.. Hodge theory of degenerations, (ii): vanishing cohomology and geometric applications, preprint, arXiv preprint arXiv:2006.03953, 2023.Google Scholar
Kerr, M and Pearlstein, G.,
Boundary components of Mumford–Tate domains
, Duke Math. J. 165 (2016), no. 4, 661–721.CrossRefGoogle Scholar
Kerr, M., Pearlstein, G. and Robles, C.,
Polarized relations on horizontal SL(2)’s
, Doc. Math. 24 (2019), 1295–1360.CrossRefGoogle Scholar
Kato, K. and Usui, S., Classifying spaces of degenerating polarized hodge structures, volume 169 of Annals of Mathematics Studies, Princeton University Press, Princeton, 2008.Google Scholar
Looijenga, E.,
A ball quotient parametrizing trigonal genus 4 curves
, Nagoya Math. J. 254 (2024), 366–378.CrossRefGoogle Scholar
McMullen, C. T.,
Braid groups and hodge theory
, Math. Ann. 355 (2013), no. 3, 893–946.CrossRefGoogle Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, vol. 34, Springer Science & Business Media, Berlin, 1994.CrossRefGoogle Scholar
Moon, H.B., Birational geometry of moduli spaces of curves of genus zero, Ph.D. thesis, Seoul National University, Aug. 2011.Google Scholar
Peters, C. A. M. and Steenbrink, J. H. M., Mixed hodge structures, volume 52 of A Series of Modern Surveys in Mathematics, Springer, Berlin, 2008.Google Scholar
Steenbrink, J.,
Intersection form for quasi-homogeneous singularities
, Compos. Math. 34 (1977), no. 2, 211–223.Google Scholar
Hoon, K. Y. and Bom, M. H.,
Moduli spaces of weighted pointed stable rational curves via GIT
, Osaka J. Math. 48 (2011), no. 4, 1115–1140.Google Scholar