Contractions

Note that for stable I-marked curves to exist, we must have |I| > 3. Let (C, λ) be a stable I-marked curve over a schemeS. Consider a subsetI⁰ ⊂I with |I⁰|>3.

Definition 1.14. Let (C⁰, λ⁰) be an I-marked curve over S, together with a morphism f: (C, λ) → (C⁰, λ⁰). We call C⁰ a contraction of C with respect to I⁰ if the pair (C⁰, λ⁰|_I⁰) is a stable I⁰-marked curve.

Intuitively, the contraction is obtained by viewingC as anI⁰-marked curve and then (on geometric fibers) contracting irreducible components containing fewer than three special points until one obtains a stable I⁰-marked curve. This is indeed the definition given by Knudsen in [9], where contractions are only explicitly defined whenI⁰ =Ir{i}

for somei∈I. The following proposition shows that the two definitions are equivalent.

Proposition 1.15. LetS= Speckfor an algebraically closed fieldk, and letI⁰ :=Ir{i}

for a fixed i ∈ I. Let (C, λ) and (C⁰, λ⁰) be I-marked curves, and suppose that (C, λ) and(C⁰, λ⁰|_I⁰)are stable. Consider a morphism f: C →C⁰ of schemes overS satisfying f◦λ_i =λ⁰_i for all i∈I. Then C⁰ is a contraction if and only if

(a) C is stable as an I⁰-marked curve and f is an isomorphism, or

(b) C is not stable as an I⁰-marked curve and f sends the irreducible component Ei

of C containing the i-marked point to a point c⁰ ∈C and induces an isomorphism CrE_i →^∼ C⁰r{c⁰}.

Proof. Under our assumptions, the morphismf is a contraction if and only if it satisfies any of the equivalent conditions in Proposition1.7. The “if” direction thus follows from the fact that (a) and (b) both imply condition (b) in Proposition1.7.

For the converse, suppose f: (C, λ) → (C⁰, λ⁰) is a contraction. If C is stable as an I⁰-marked curve, then f is an isomorphism by Proposition 1.12, so (i) holds. If C is not stable, we must show that the closed subset Z ⊂ C⁰ as defined in (1.1) consists exclusively of the point c⁰_i := λ⁰_i(S) and that f⁻¹(c⁰_i) = E_i. Let c⁰ ∈ Z and consider the set E of irreducible components of Crf⁻¹(c⁰) which have non-empty intersection with f⁻¹(c⁰). By the same reasoning as in the proof of Proposition 1.12, we must have 0<|E|62.

If |E| = 2, then, by the same reasoning as in the proof of Proposition 1.12, the stability ofC implies that the fiberf⁻¹(c⁰) contains an I-marked point. We claim that

the only marked point in f⁻¹(c⁰) is the i-marked point. Indeed, in this case c⁰ ∈ C⁰ is a nodal singularity in C⁰. Since C⁰ is stable, the I⁰-marked points are smooth by definition. The fiber f⁻¹(c⁰) therefore cannot contain an I⁰-marked point, which yields the claim. The stability ofC then implies thatf⁻¹(c⁰) is irreducible and hence equal to E_i, as desired.

By similar reasoning, if|E|= 1, the fiber f⁻¹(c⁰) must contain precisely two marked points, and these correspond toiand somei⁰ ∈I⁰. The stability ofC again implies that f⁻¹(c⁰) is irreducible and hence equal to E_i.

Figure 2: Example of a contraction forI :={1,2,3,4,5} and I⁰ :={1,2,3,5}.

λ₅

λ₅ λ3, λ4

λ₁ λ2

λ₃ λ4

λ1

λ₂

Proposition 1.16. Let (C, λ) be a stable I-marked curve, and let I⁰ be a subset of I with |I⁰| > 3. There exists a contraction f: (C, λ) → (C⁰, λ⁰) of C with respect to I⁰. The tuple (C⁰, λ⁰, f) is unique up to unique isomorphism.

Proof. The existence and uniqueness of contractions whenI⁰ =Ir{i}for some i∈I is proven in [9, Proposition 2.1]. Iterating this (using that the composite of morphisms of I-marked curves is again such a morphism) and noting that the result is independent of the order in which we forget sections [1, Lemma 10.6.10], yields the desired contraction for general I⁰.

Let (C⁰, λ⁰) be the contraction of (C, λ) with respect toI⁰. For each i∈I, we denote the image of the i-marked section by Zi ⊂ C⁰. SinceC⁰ is separated over S, each Zi is closed in C⁰. Consider

Z_(IrI⁰₎ := [

i∈(IrI⁰)

Z_i.

The following property of contractions will be used repeatedly in later sections.

Proposition 1.17. The contraction morphism f: C→C⁰ induces an isomorphism Crf⁻¹(Z_(IrI⁰₎)→^∼ C⁰rZ_(IrI⁰₎.

Proof. Lets ∈S, and let c⁰ ∈ C_s⁰ be a point with dimf_s⁻¹(c⁰) = 1. We consider the set of irreducible components of C_s rf_s⁻¹(c⁰) which intersect f_s⁻¹(c⁰) and apply the same argument as in the proof of Proposition1.15 to deduce thatf_s⁻¹(c⁰) contains an (IrI⁰ )-marked point. It follows that Z ⊂C⁰ as defined in (1.1) is contained inZ_(IrI⁰₎, and the proposition follows by the definition of a morphism of I-marked curves.

Irreducible components of stable I-marked curves

The following lemma and proposition give a nice consequence of the existence of con-tractions. Let (C, λ) be a stable I-marked curve over Speck, wherek is a field, and fix an algebraic closure k of k.

Lemma 1.18. Every irreducible component of C is geometrically irreducible.

Proof. We proceed by induction on |I|. If |I| = 3, then it follows directly from the definition of stability that the base change C_k must be isomorphic to P¹_k; hence C is geometrically irreducible. Suppose |I| =n >3. Let i∈ I and consider the contraction C⁰ of C with respect to I⁰ := I r {i}. Let Z ⊂ C⁰ be as defined in (1.1). Since

|I⁰|=n−1, it follows from the induction hypothesis that every irreducible component of C⁰ is geometrically irreducible. If Z =∅, then the contraction morphismf: C →C⁰ is an isomorphism by definition, so we deduce the same for the irreducible components of C. Otherwise, the base change Z_k must consist of a single point, whose image in C⁰ we denote by c⁰ (so that Z = {c⁰}). Then f⁻¹(c⁰)_k is irreducible, so f⁻¹(c⁰) is geometrically irreducible. By definition, the morphism f induces an isomorphism Crf⁻¹(c⁰)→^∼ C⁰ r{c⁰}. The induction hypothesis then implies that every irreducible component ofCrf⁻¹(c⁰) is geometrically irreducible. SinceC = Crf⁻¹(c⁰)

∪f⁻¹(c⁰), this proves the lemma.

Proposition 1.19. Let E ⊂C be an irreducible component. Then E ∼=P¹k.

Proof. If E contains an I-marked point, then it is isomorphic to P¹k because any con-nected smooth projective curve of genus 0 over Speck possessing a k-rational point is isomorphic to P¹k. Suppose E contains no I-marked points. The base change E_k is irreducible by Lemma 1.18 and also contains no I-marked points. It follows that E_k contains as least three singular points. This implies that C_k rE_k has at least three connected components. Since C is stable, we can choose a marked point on each of these components. The chosen points correspond to some I⁰ := {i, j, k} ⊂ I. Let C⁰ denote the contraction ofC with respect toI⁰. ThenC⁰ contains a marked point and is thus isomorphic toP¹k. The contraction morphismf: C→C⁰ then induces a morphism E →C⁰ ∼=P¹k. This becomes an isomorphism after base change to k and is hence itself an isomorphism.

Remark. It follows from Proposition 1.19 that we may replace every instance of the geometric fiber in Definitions 1.1 and 1.14 by the (scheme-theoretic) fiber, and we will often do so in the remainder of the text.