Seminar: Stacks and Construction of Mg
Winter term 2016/17
Time: Thursday 4-6pm
Room: Room 0.008
Contact: Ulrike Rieß
Professor: Prof. Dr. Daniel Huybrechts
First session for introduction (& organization): 20.10.2016
First regular talk: 27.10.2016
If you want to participate in the seminar, please write an email to Ulrike Rieß (uriess
(at)math.uni-bonn.de).
Program
The aim of this seminar is to understand the construction of Mg , which is the compactification of the moduli space of curves of genus g (as constructed in [DM72]). To this
goal, we introduce the concept of stacks and discuss known moduli functors, properties
of curves, the construction of Mg itself, and stable curves. The general approach will
mainly follow [Edi00].
This version of the program contains the final list of talks, but some details might
still change and there will be additional references, where needed.
Talk 1. Classical moduli functors I. (27.10. - Orlando Marigliano)
Introduce the concept of a representable functor (see [FGI+ 05, Definition 2.1]) and of
a universal object for a functor (see [FGI+ 05, Definition 2.2]). Use the Yoneda Lemma
to to deduce that a functor is representable if and only if it admits a universal object
[FGI+ 05, Proposition 2.3].
For a set of objects and a notion what families of these objects are, define the
associated moduli functor
F : (SchC )op → (Sets),
F(T ) := {families over T }/ ∼,
for a “suitable” equivalence relation ∼. Define its moduli space as a representing object,
and observe that the set F(Spec(C)) corresponds to the rational points of the moduli
space on the one hand, and to the parametrized objects (possibly up to some relation)
on the other hand. Define a universal family as universal object for the functor and show
that its fibre over a point in the moduli space, is just the object the point represents.
As a first step define for a fixed complex variety X the moduli functor of its points:
Hom(_, X). Say that it makes sense to speak of Hom(T, X) as a family of points in
X parametrized by T and mention all of these introduced objects for this functor.
Discuss the Grassmannian as a first serious example: Define the functor for the
Grassmannian, state that its rational points are in bijection to the quotients, and that
the fibres of a universal family would correspond to the respective quotient. Present
the construction of the Grassmannian (as in [HL10, Example 2.2.2]).
End with writing down a functor for the moduli of smooth curves of genus g (see
[Edi00, p. 86]), and say why no scheme can be a fine moduli space for it.
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Talk 2. Classical moduli functors II. (3.11. - Tobias Töpfer)
Define the functor GrassS (V, r) as in [HL10, Example 2.2.3], introduce the functor
QuotX/S (H, P ) (as in [HL10, p. 42]), and define an analogous functor for HilbP (X)
(compare [HL10, p. 44]). Show that GrassS (V, r) is representable ([HL10, Example
2.2.3]). Present the construction of QuotX/S (H, P ) (see [HL10, Theorem 2.2.4]) and
deduce HilbP (X) (as on page 44 of [HL10]).
Talk 3. Properties of smooth curves. (10.11. - Thorsten Beckmann)
Recall that for a smooth curve it is equivalent to be proper or projective.
Show that for a smooth projective curve C with genus g ≥ 2 there are only finitely
many automorphisms: | Aut(C)| < ∞ (see e.g. [ACGH85, Exercises on p.45] or [Ros55]).
Prove that for smooth projective curves of genus g ≥ 2 the line bundles ωC⊗n are very
ample for n ≥ 3 (see e.g. [Har77, Corollary IV.3.2 + Example IV.1.3.3]).
Show that for a flat family π : C → S of smooth curves of genus g ≥ 2, the sheaf
π∗ (ωC⊗n|S ) is a vector bundle of rank (2n − 1)(g − 1) (this uses Riemann–Roch, Serre
duality, and the following base change theorem: [FGI+ 05, Theorem 5.10.(3)]).
Talk 4. Étale topology. (17.11. - Fabian Koch)
Start with recalling basic properties of étale morphisms. Mention that open immersions
are étale, étale morphisms are flat, and thus étale surjective morphisms between affine
schemes induce faithfully flat homomorphisms of rings (compare e.g. [Liu06, Corollary
I.2.20]).
Introduce the notion of a Grothendieck topology [FGI+ 05, Definition 2.24]. Define
the Zarisky topology for schemes and the étale topology for schemes as in [FGI+ 05,
Examples 2.30 + 2.29 + 2.31] (you can take [FGI+ 05, Examples 2.26 + 2.27] as the
motivation).
Define when a functor is a sheaf with respect to some Grothendieck topology (see
[FGI+ 05, Section 2.3.3]), and observe that the functor of points for a scheme is a sheaf
in the Zariski topology. Furthermore, show that the functor of points for a scheme is
a sheaf in the étale topology (e.g. as in [Zom14], more details can be found in [GW10,
Proposition 14.74] or [FGI+ 05, Theorem 2.55]). Furthermore state [Zom14, Theorem
1.6] for the property “proper” (compare also [GW10, Proposition 14.51.(5)]) and prove
it if you have time.
Talk 5 + 6. Stacks I + II. (24.11. + 1.12. - Stefano Ariotta, Jonathan Gruner)
Definition of a category fibred in groupoids (CFG) [BCE+ 07, Definition 2.19], CFG
associated to a scheme [BCE+ 07, Example 2.1], and CFG associated to a functor
[BCE+ 07, Example 2.4]. Introduce the moduli groupoid Mg of smooth curves [BCE+ 07,
Example 2.2]. Remark that the image of the moduli functor for curves is not isomorphic
to the moduli CFG for curves. Observe that for any given base T , families of smooth
proper curves over T (together with a choice of pullbacks along arbitrary morphisms)
are in bijection with morphisms T → Mg .
Define principal G-bundles and G-equivariant maps [BCE+ 07, Example 1.1.B, may
add things from p. 31, 32] and [X/G] quotient groupoids for a scheme [BCE+ 07, Example 2.6].
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Define morphisms and isomorphisms of CFGs [BCE+ 07, p. 33, 34]. Define the morphism X → [X/G] [BCE+ 07, Example 2.9.4].
Discuss the functor (Sch) → (CFGs) and state that it embeds (Sch) as a full subcategory (compare “weak 2-Yoneda lemma” [FGI+ 05, p. 59] - state what the maps in both
directions are). Deduce that two schemes are isomorphic if and only if their associated
CFGs are.
Construct the fibre product in the category of CFGs (see [BCE+ 07, Section 2.5]).
Be very carefull about the necessary 2-morphisms here. State the universal property
[BCE+ 07, Remark 2.31].
Prove that the fibre product satisfies this property (and even the stronger “strict
universal property” [BCE+ 07, p. 46]).
Definition of a stack (see [Edi00, Definition 2.4], [BCE+ 07, Definition 4.3]). Show
that the CFG associated to a functor is a stack if and only if the functor is a sheaf in
the étale topology. Deduce that the CFG associated to a scheme is a stack.
Show that the fibre product of stacks is again a stack (see [BCE+ 07, p. 77]).
Show that under certain conditions the quotient groupoid of a scheme is a stack
(compare [Edi00, Proposition 2.2] - use [BCE+ 07, Corollary A.17], you can prove this,
if you have time).
—— The session on 8.12 is earlier (12h - 14h) and in Room 1.007 ——
Talk 7. Deligne–Mumford stacks I. (8.12.:12-14h, Room 1.007 - Tim Bülles)
Define representable morphisms between CFGs for stacks (see [BCE+ 07, Definition
5.1]). Note that a morphism between the CFGs associated to a scheme is representable
([BCE+ 07, Example 5.2.1]). Define certain properties of representable morphisms
([BCE+ 07, Definition 5.3], state parts of [BCE+ 07, Proposition 5.5]: for surjective,
quasi-compact, separated, locally of finite type, proper, flat, open embedding, closed
embedding, isomorphism, étale). Show that is G is smooth, and affine, then X → [X/G]
is a representable morphism which is smooth and surjective [Edi00, Example 2.10 +
Example 2.12].
State and prove [BCE+ 07, Proposition 5.12].
Define a Deligne–Mumford stack: [BCE+ 07, Definition 5.14]. Note that the stack
associated to a scheme is a Deligne–Mumford stack ([BCE+ 07, Example 5.16]).
State [Edi00, Corollary 2.1]. You can prove this, if you have time.
Talk 8. Deligne–Mumford stacks II. (15.12. - Simone Fabrizzi)
Show that quotient stacks are Deligne–Mumford stacks if the group acts with finite
and reduced stabilizers of geometric points.
This talk should mainly follow the exposition in [Edi00].
—— No session on 22.12. ——
Talk 9. Construction of Mg . (12.1. - João Lourenço)
Show that Mg is isomorphic to a certain quotient stack, and that this quotient is in
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fact a Deligne–Mumford stack. (Proof: Use [Edi00] and ignore the words “stable curve”
and Mg , and just use the statements for “families of smooth curves” and Mg instead).
Talk 10. Stable curves. (19.1. - Isabelle Große-Brauckmann)
Definition of stable curves. Prove that the automorphism group of a stable curve
is finite. Introduce the canonical sheaf ωC for stable curves. (One can follow the
exposition from [Edi00]). State and prove [DM72, Theorem 1.2].
Talk 11. Construction of Mg . (26.1. - Mafalda Santos)
Introduce the moduli stack for stable curves and prove that it is isomorphic to a
certain quotient stack. Deduce that it is a Deligne–Mumford stack. (One can follow
the exposition from [Edi00]).
Talk 12. Properness and smoothness of Mg . (2.2. - Emma Brakkee)
See [DM72, Theorem 5.2] and [Edi00, Section 3.2] (only state/sketch the stable degeneration result).
Talk 13. Irreducibility of Mg . (9.2. - David Linus Hamann)
Proof as in [Edi00] or [DM72].
References
[ACGH85] Enrico Arbarello, Maurizio Cornalba, Phillip A. Griffiths, and Joseph D. Harris.
Geometry of algebraic curves. Volume I, volume 267 of Grundlehren der mathematischen
Wissenschaften. Springer-Verlag, 1985.
[BCE+ 07] Kai Behrend, Brian Conrad, Dan Edidin, Barbara Fantechi, William Fulton,
Lothar Göttsche, and Andrew Kresch. Algebraic stacks. unpublished - available at:
http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1,
2007.
[DM72]
Pierre Deligne and David Mumford. The irreducibility of the space of curves of a given
genus. Matematika, Moskva, 16(3):13–53, 1972.
[Edi00]
Dan Edidin. Notes on the construction of the moduli space of curves. In Recent progress in
intersection theory. Based on the international conference on intersection theory, Bologna,
Italy, December 1997, pages 85–113. Boston, MA: Birkhäuser, 2000.
[FGI+ 05] Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and
Angelo Vistoli. Fundamental Algebraic Geometry: Grothendieck’s FGA Explained, volume
123 of Mathematical Surveys and Monographs. American Mathematical Society (AMS),
2005.
[GW10]
Ulrich Görtz and Torsten Wedhorn. Algebraic geometry I. Wiesbaden: Vieweg+Teubner,
2010.
[Har77]
Robin Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics. Springer, New
York, 1977.
[HL10]
Daniel Huybrechts and Manfred Lehn. The geometry of moduli spaces of sheaves. 2nd ed.
Cambridge: Cambridge University Press, 2010.
[Liu06]
Qing Liu. Algebraic Geometry and Arithmetic Curves, volume 6 of Oxford Graduate Texts
in Mathematics. Oxford University Press, 2006.
[Ros55]
Maxwell Rosenlicht. Automorphisms of function fields. Trans. Am. Math. Soc., 79:1–11,
1955.
[Zom14]
Wouter Zomervrucht. Descent on the étale site. Lecturenotes available at:
http://pub.math.leidenuniv.nl/~zomervruchtw/docs/etdesc.pdf, 2014.
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(Last updated 29.11.2016)