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12 1. A gentle introduction to moduli spaces of curvesAdmittedly, the definition of a Deligne–Mumford stack is rather technical. To presentit here would take us too far afield from our goal and may even obscure the geometricnature of moduli spaces. However, one will not go too far wrong thinking of a smoothDeligne–Mumford stack as a complex orbifold. The stack structure takes care of the extrabookkeeping required to deal with curves with non-trivial automorphisms. Essentially,this will allow us to treat the moduli space as smooth and ensure that there is a universalfamily over it. The reader interested in discovering more on stacks is encouraged toconsult [14] and the references contained therein.Those troublesome curves with non-trivial automorphisms have led us to consider modulistacks of pointed curves. However, there is still one outstanding issue which needs to be addressed— M g,n is not compact. To see this, observe that when two marked points on a pointedcurve approach each other, then the corresponding limit does not exist in M g,n . Of course, thereare numerous advantages in working with a compact space. And although there are variousways to compactify the moduli space, it is becoming increasingly clear that the most profitableis to use what is known as the Deligne–Mumford compactification. One of its virtues is that itis modular — in other words, it is a moduli space for a well-behaved, easy to describe, class ofcurves. In fact, all we need to do is gently relax our smoothness condition. Thus, we define theDeligne–Mumford compactification of the moduli space{M g,n = (C, p 1 , p 2 , . . . , p n )∣}/C is a stable algebraic curve of genus g with∼n distinct smooth points p 1 , p 2 , . . . , p nwhere (C, p 1 , p 2 , . . . , p n ) ∼ (D, q 1 , q 2 , . . . , q n ) if and only if there exists an isomorphism from Cto D which sends p k to q k for all k. Here, an algebraic curve is called stable if it has at worstnodal singularities and a finite automorphism group. The practical interpretation of this lattercondition is that every rational component of the curve must have at least three special points,where a special point refers to a node or a marked point. It should be clear that M g,n ⊆ M g,n ,and we refer to M g,n \ M g,n as the boundary divisor. Theorem 1.1 can be extended to thecompactification as follows.Theorem 1.2. There exists a finite cover ˜M g,n → M g,n such that ˜M g,n is a smooth manifold. Furthermore,the boundary divisor lifts to a union of codimension two submanifolds intersecting transversally.The question still remains as to which curve arises in the limit when two marked points approacheach other. To see what the correct answer should be, consider a family of genus gcurves with n marked points φ : X → B, where B is smooth and of complex dimension 1. Themarked points give rise to n sections which we denote by σ 1 , σ 2 , . . . , σ n : B → X. Supposethat all fibres are stable pointed curves apart from over the point b where the sections σ i andσ j intersect. In order to obtain a family of stable curves, one simply needs to blow up the surfaceX at the point σ i (b) = σ j (b) to obtain π : ˜X → X. The new family of curves is given by
1.1. Moduli spaces of curves 13˜φ : ˜X → B where ˜φ = φ ◦ π. Note that the fibres are unchanged away from b, whereas thenew fibre over b consists of the old fibre plus the exceptional divisor obtained from blowing up.Furthermore, the points σ i (b) and σ j (b) now lie on the exceptional divisor and are distinct aslong as the sections σ i and σ j intersected transversally. Therefore, in the limit when two markedpoints approach each other, a CP 1 bubbles off, containing these two marked points. This is aparticular instance of the more general process known as stable reduction. 4Theorem 1.3 (Stable reduction). Let b be a point on a smooth curve B and suppose that X is a family ofstable curves over B \ {b}. Then after a sequence of blow-ups and blow-downs and passing to a branchedcover of B, one can obtain a new family where all fibres are stable curves. Furthermore, the fibre over b inthis new family is uniquely determined.More complicated stable pointed curves arise from more complicated limits, such as the exampleshown in the following diagram. 5236781495We note now the important foundational result thatdim R M g,n = 6g − 6 + 2n,a calculation which dates back to Riemann. Later in this chapter, we will see how the uncompactifiedmoduli space M g,n actually arises as the quotient of an open ball of real dimension6g − 6 + 2n by a properly discontinuous group action.Natural morphismsOne interesting and useful aspect of moduli spaces of pointed curves is the interplay betweenthem. As a simple example of this phenomenon, note that if g ≤ g ′ and n ≤ n ′ , then M g,n canbe considered a subvariety of M g ′ ,n′. The following natural morphisms between moduli spacesof curves are of particular importance.4 For further details on stable reduction, one need look no further than [22]. In fact, the book is an excellent referenceon the algebraic geometry of moduli spaces of curves in general.5 Recall that the singularities of a stable pointed curve must be nodal and, hence, locally look like xy = 0 at the origin.In order to represent such a singularity on a two-dimensional page, one usually resorts to drawing the two curves aspinched or tangent, both of which are misleading.
Page 1: Intersection <stro
Page 4 and 5: iiDeclarationThis is to certify tha
Page 6 and 7: ivAcknowledgementsFirst and foremos
Page 8 and 9: viContentsVolumes and symplectic re
Page 10 and 11: viiiContents
Page 12 and 13: 2 0. Overviewwhich is far more trac
Page 14 and 15: 4 0. Overviewcurves. In this thesis
Page 16 and 17: 6 0. Overvieware defined to be grap
Page 18 and 19: 8 0. OverviewNote that the space of
Page 20 and 21: 10 1. A gentle introduction to modu
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Page 39 and 40: 1.3. Moduli spaces of hyperbolic su
Page 41 and 42: 1.4. Volumes of moduli spaces 31The
Page 43 and 44: 1.4. Volumes of moduli spaces 33vol
Page 45 and 46: 1.4. Volumes of moduli spaces 35In
Page 47 and 48: 1.4. Volumes of moduli spaces 37reg
Page 49 and 50: 1.4. Volumes of moduli spaces 39Pro
Page 51 and 52: Chapter 2Weil-Petersson volume rela
Page 53 and 54: 2.1. Volume polynomial relations 43
Page 55 and 56: 2.2. Proofs via algebraic geometry
Page 57 and 58: 2.2. Proofs via algebraic geometry
Page 59 and 60: 2.3. Proofs via Mirzakhani’s recu
Page 69 and 70: 2.4. Applications and extensions 59
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2.4. Applications and extensions 63
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Chapter 3A new approach to Kontsevi
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3.1. Kontsevich’s combinatorial f
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3.2. Hyperbolic surfaces and ribbon
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3.3. Hyperbolic surfaces with long
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3.4. Calculating the combinatorial
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Chapter 4Concluding remarksIn this
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Bibliography[1] ABIKOFF, W. The rea
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Bibliography 103[27] KRONHEIMER, P.
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Bibliography 105[58] WOLPERT, S. On
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Appendix AWeil-Petersson volumesA.1
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A.2. Program for genus 0 Weil-Peter
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