Seminar: Stacks and Construction of M g Winter term 2016/17 Time: Thursday 4-6pm Room: Room 0.008

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1 Seminar: Stacks and Construction of M g Winter term 2016/17 Time: Thursday 4-6pm Room: Room Contact: Ulrike Rieß Professor: Prof. Dr. Daniel Huybrechts First session for introduction (& organization): First regular talk: If you want to participate in the seminar, please write an to Ulrike Rieß (uriess (at)math.uni-bonn.de). Program The aim of this seminar is to understand the construction of M g, 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 M g 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. ( 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 : (Sch C ) 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. 1

2 2 Talk 2. Classical moduli functors II. ( Tobias Töpfer) Define the functor Grass S (V, r) as in [HL10, Example 2.2.3], introduce the functor Quot X/S (H, P ) (as in [HL10, p. 42]), and define an analogous functor for Hilb P (X) (compare [HL10, p. 44]). Show that Grass S (V, r) is representable ([HL10, Example 2.2.3]). Present the construction of Quot X/S (H, P ) (see [HL10, Theorem 2.2.4]) and deduce Hilb P (X) (as on page 44 of [HL10]). Talk 3. Properties of smooth curves. ( 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 ω n C are very ample for n 3 (see e.g. [Har77, Corollary IV Example IV.1.3.3]). Show that for a flat family π : C S of smooth curves of genus g 2, the sheaf π (ω n C 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. ( 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 ] (you can take [FGI + 05, Examples ] 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 (5)]) and prove it if you have time. Talk Stacks I + II. ( 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 M g 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 M g. 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].

3 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 Talk 7. Deligne Mumford stacks I. (8.12.:12-14h, Room 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 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. ( 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 Talk 9. Construction of M g. ( João Lourenço) Show that M g is isomorphic to a certain quotient stack, and that this quotient is in 3

4 4 fact a Deligne Mumford stack. (Proof: Use [Edi00] and ignore the words stable curve and M g, and just use the statements for families of smooth curves and M g instead). Talk 10. Stable curves. ( Isabelle Große-Brauckmann) Definition of stable curves. Prove that the automorphism group of a stable curve is finite. Introduce the canonical sheaf ω C for stable curves. (One can follow the exposition from [Edi00]). State and prove [DM72, Theorem 1.2]. Talk 11. Construction of M g. ( 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 M g. ( Emma Brakkee) See [DM72, Theorem 5.2] and [Edi00, Section 3.2] (only state/sketch the stable degeneration result). Talk 13. Irreducibility of M g. ( 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, [BCE + 07] Kai Behrend, Brian Conrad, Dan Edidin, Barbara Fantechi, William Fulton, Lothar Göttsche, and Andrew Kresch. Algebraic stacks. unpublished - available at: [DM72] Pierre Deligne and David Mumford. The irreducibility of the space of curves of a given genus. Matematika, Moskva, 16(3):13 53, [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 Boston, MA: Birkhäuser, [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), [GW10] Ulrich Görtz and Torsten Wedhorn. Algebraic geometry I. Wiesbaden: Vieweg+Teubner, [Har77] Robin Hartshorne. Algebraic Geometry. Graduate Texts in Mathematics. Springer, New York, [HL10] Daniel Huybrechts and Manfred Lehn. The geometry of moduli spaces of sheaves. 2nd ed. [Liu06] Cambridge: Cambridge University Press, Qing Liu. Algebraic Geometry and Arithmetic Curves, volume 6 of Oxford Graduate Texts in Mathematics. Oxford University Press, [Ros55] Maxwell Rosenlicht. Automorphisms of function fields. Trans. Am. Math. Soc., 79:1 11, [Zom14] Wouter Zomervrucht. Descent on the étale site. Lecturenotes available at:

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