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Birational models of π“œ2,2 arising as moduli of curves with nonspecial divisors

Drew Johnson and Alexander Polishchuk
From the journal Advances in Geometry

Abstract

We study birational projective models of π“œ2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves π“œ2,2(𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M2,2(𝓩).

MSC 2010: 14H10; 14D23

  1. Communicated by: R. Cavalieri

References

[1] J. Alper, M. Fedorchuk, D. I. Smyth, F. Van der Wyck, Second flip in the Hassett–Keel programm: a local description. Compositio Math. 153 (2017), 1547–1583. MR3705268 Zbl 1403.14039 Search in Google Scholar

[2] D. Eisenbud, Commutative algebra. Springer 1995. MR1322960 Zbl 0819.13001 Search in Google Scholar

[3] M. Fedorchuk, D. I. Smyth, Alternate compactifications of moduli spaces of curves. In: Handbook of moduli. Vol. I, volume 24 of Adv. Lect. Math. (ALM), 331–413, Int. Press, Somerville, MA 2013. MR3184168 Zbl 1322.14048 Search in Google Scholar

[4] G.-M. Greuel, On deformation of curves and a formula of Deligne. In: Algebraic geometry (La RΓ‘bida, 1981), volume 961 of Lecture Notes in Math., 141–168, Springer 1982. MR708332 Zbl 0509.14031 Search in Google Scholar

[5] A. Polishchuk, Moduli of curves as moduli of A∞-structures. Duke Math. J. 166 (2017), 2871–2924. MR3712167 Zbl 1401.14099 arXiv:1312.4636 [math.AG] Search in Google Scholar

[6] A. Polishchuk, Moduli of curves, GrΓΆbner bases, and the Krichever map. Adv. Math. 305 (2017), 682–756. MR3570146 Zbl 1356.14023 Search in Google Scholar

[7] A. Polishchuk, Moduli spaces of nonspecial pointed curves of arithmetic genus 1. Math. Ann. 369 (2017), 1021–1060. MR3713534 Zbl 1386.14109 Search in Google Scholar

[8] A. Polishchuk, Contracting the Weierstrass locus to a point. In: String-Math 2016, volume 98 of Proc. Sympos. Pure Math., 241–257, Amer. Math. Soc. 2018. MR3821756 Search in Google Scholar

[9] D. I. Smyth, Towards a classification of modular compactifications of π“œg,n. Invent. Math. 192 (2013), 459–503. MR3044128 Zbl 1277.14023 Search in Google Scholar

Received: 2018-09-30
Revised: 2019-01-30
Published Online: 2021-01-22
Published in Print: 2021-01-27

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