21,243 reputation
26191
bio website homepage.sns.it/vistoli
location Italy
age 57
visits member for 5 years, 5 months
seen Jun 22 at 17:56

My name is Angelo Vistoli. I do algebraic geometry, mostly moduli theory.

Normally I don't answer questions from anonymous users. Lately I have lost interest, and hardly answered any question at all.


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comment Do canonical stacks exist over Spec(Z)?
No, I don’t think that $S$ being regular is necessary to conclude that $U/H$ is smooth; flatness follows from the fact that $H$ is tame, and then smoothness can be checked on the fibers. The fact that $S$ is regular is used in the last step, when you normalize the fiber product $U \times_S V$, and use purity of the branch locus to conclude that it is étale over$U$ and $V$.
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revised Do canonical stacks exist over Spec(Z)?
added 4 characters in body
Feb
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answered Do canonical stacks exist over Spec(Z)?
Dec
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comment What is the expected dimension of the Zariski closure of the rational points on the moduli space of curves?
This follows from the Casnati-Ekedahl description of coverings of degrees 4 and 5 (see G. Casnati, T. Ekedahl: Covers of algebraic varieties I. A general structure theorem, covers of degree 3,4 and Enriques' surfaces. J. Algebraic Geom., 5 (1996), pp. 439-460, and G. Casnati: Covers of algebraic varieties II. Covers of degree 5 and construction of surfaces. J. Algebraic Geom., 5 (1996) pp 461-477) together with the description of the stack of globally generated vector bundles on P^1 in M. Bolognesi, A. Vistoli, Stacks of trigonal curves, Trans. Amer. Math. Soc., 364 (2012), 3365–3393.
Dec
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comment What is the expected dimension of the Zariski closure of the rational points on the moduli space of curves?
I believe that one can show that the stack of curves with a map of degree $d$ to a $\mathbb P^1$ is dominated by a rational variety for all $d \le 5$. This would give a lower bound $2g+5$ for the dimension of the closure.
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