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The Stacks project

55.9 The geometry of a regular model

In this section we describe the geometry of a proper regular model X of a smooth projective curve C over K with H^0(C, \mathcal{O}_ C) = K.

Lemma 55.9.1. Let X be a regular model of a smooth curve C over K.

  1. the special fibre X_ k is an effective Cartier divisor on X,

  2. each irreducible component C_ i of X_ k is an effective Cartier divisor on X,

  3. X_ k = \sum m_ i C_ i (sum of effective Cartier divisors) where m_ i is the multiplicity of C_ i in X_ k,

  4. \mathcal{O}_ X(X_ k) \cong \mathcal{O}_ X.

Proof. Recall that R is a discrete valuation ring with uniformizer \pi and residue field k = R/(\pi ). Because X \to \mathop{\mathrm{Spec}}(R) is flat, the element \pi is a nonzerodivisor affine locally on X (see More on Algebra, Lemma 15.22.11). Thus if U = \mathop{\mathrm{Spec}}(A) \subset X is an affine open, then

X_ K \cap U = U_ k = \mathop{\mathrm{Spec}}(A \otimes _ R k) = \mathop{\mathrm{Spec}}(A/\pi A)

and \pi is a nonzerodivisor in A. Hence X_ k = V(\pi ) is an effective Cartier divisor by Divisors, Lemma 31.13.2. Hence (1) is true.

The discussion above shows that the pair (\mathcal{O}_ X(X_ k), 1) is isomorphic to the pair (\mathcal{O}_ X, \pi ) which proves (4).

By Divisors, Lemma 31.15.11 there exist pairwise distinct integral effective Cartier divisors D_ i \subset X and integers a_ i \geq 0 such that X_ k = \sum a_ i D_ i. We can throw out those divisors D_ i such that a_ i = 0. Then it is clear (from the definition of addition of effective Cartier divisors) that X_ k = \bigcup D_ i set theoretically. Thus C_ i = D_ i are the irreducible components of X_ k which proves (2). Let \xi _ i be the generic point of C_ i. Then \mathcal{O}_{X, \xi _ i} is a discrete valuation ring (Divisors, Lemma 31.15.4). The uniformizer \pi _ i \in \mathcal{O}_{X, \xi _ i} is a local equation for C_ i and the image of \pi is a local equation for X_ k. Since X_ k = \sum a_ i C_ i we see that \pi and \pi _ i^{a_ i} generate the same ideal in \mathcal{O}_{X, \xi _ i}. On the other hand, the multiplicity of C_ i in X_ k is

m_ i = \text{length}_{\mathcal{O}_{C_ i, \xi _ i}} \mathcal{O}_{X_ k, \xi _ i} = \text{length}_{\mathcal{O}_{C_ i, \xi _ i}} \mathcal{O}_{X, \xi _ i}/(\pi ) = \text{length}_{\mathcal{O}_{C_ i, \xi _ i}} \mathcal{O}_{X, \xi _ i}/(\pi _ i^{a_ i}) = a_ i

See Chow Homology, Definition 42.9.2. Thus a_ i = m_ i and (3) is proved. \square

Lemma 55.9.2. Let X be a regular model of a smooth curve C over K. Then

  1. X \to \mathop{\mathrm{Spec}}(R) is a Gorenstein morphism of relative dimension 1,

  2. each of the irreducible components C_ i of X_ k is Gorenstein.

Proof. Since X \to \mathop{\mathrm{Spec}}(R) is flat, to prove (1) it suffices to show that the fibres are Gorenstein (Duality for Schemes, Lemma 48.25.3). The generic fibre is a smooth curve, which is regular and hence Gorenstein (Duality for Schemes, Lemma 48.24.3). For the special fibre X_ k we use that it is an effective Cartier divisor on a regular (hence Gorenstein) scheme and hence Gorenstein for example by Dualizing Complexes, Lemma 47.21.6. The curves C_ i are Gorenstein by the same argument. \square

Situation 55.9.3. Let R be a discrete valuation ring with fraction field K, residue field k, and uniformizer \pi . Let C be a smooth projective curve over K with H^0(C, \mathcal{O}_ C) = K. Let X be a regular proper model of C. Let C_1, \ldots , C_ n be the irreducible components of the special fibre X_ k. Write X_ k = \sum m_ i C_ i as in Lemma 55.9.1.

Proof. Consequence of More on Morphisms, Lemma 37.53.6. \square

Lemma 55.9.5. In Situation 55.9.3 there is an exact sequence

0 \to \mathbf{Z} \to \mathbf{Z}^{\oplus n} \to \mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (C) \to 0

where the first map sends 1 to (m_1, \ldots , m_ n) and the second maps sends the ith basis vector to \mathcal{O}_ X(C_ i).

Proof. Observe that C \subset X is an open subscheme. The restriction map \mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (C) is surjective by Divisors, Lemma 31.28.3. Let \mathcal{L} be an invertible \mathcal{O}_ X-module such that there is an isomorphism s : \mathcal{O}_ C \to \mathcal{L}|_ C. Then s is a regular meromorphic section of \mathcal{L} and we see that \text{div}_\mathcal {L}(s) = \sum a_ i C_ i for some a_ i \in \mathbf{Z} (Divisors, Definition 31.27.4). By Divisors, Lemma 31.27.6 (and the fact that X is normal) we conclude that \mathcal{L} = \mathcal{O}_ X(\sum a_ iC_ i). Finally, suppose that \mathcal{O}_ X(\sum a_ i C_ i) \cong \mathcal{O}_ X. Then there exists an element g of the function field of X with \text{div}_ X(g) = \sum a_ i C_ i. In particular the rational function g has no zeros or poles on the generic fibre C of X. Since C is a normal scheme this implies g \in H^0(C, \mathcal{O}_ C) = K. Thus g = \pi ^ a u for some a \in \mathbf{Z} and u \in R^*. We conclude that \text{div}_ X(g) = a \sum m_ i C_ i and the proof is complete. \square

In Situation 55.9.3 for every invertible \mathcal{O}_ X-module \mathcal{L} and every i we get an integer

\deg (\mathcal{L}|_{C_ i}) = \chi (C_ i, \mathcal{L}|_{C_ i}) - \chi (C_ i, \mathcal{O}_{C_ i})

by taking the degree of the restriction of \mathcal{L} to C_ i relative to the ground field k1 as in Varieties, Section 33.44.

Lemma 55.9.6. In Situation 55.9.3 given \mathcal{L} an invertible \mathcal{O}_ X-module and a = (a_1, \ldots , a_ n) \in \mathbf{Z}^{\oplus n} we define

\langle a, \mathcal{L} \rangle = \sum a_ i\deg (\mathcal{L}|_{C_ i})

Then \langle , \rangle is bilinear and for b = (b_1, \ldots , b_ n) \in \mathbf{Z}^{\oplus n} we have

\left\langle a, \mathcal{O}_ X(\sum b_ i C_ i) \right\rangle = \left\langle b, \mathcal{O}_ X(\sum a_ i C_ i) \right\rangle

Proof. Bilinearity is immediate from the definition and Varieties, Lemma 33.44.7. To prove symmetry it suffices to assume a and b are standard basis vectors in \mathbf{Z}^{\oplus n}. Hence it suffices to prove that

\deg (\mathcal{O}_ X(C_ j)|_{C_ i}) = \deg (\mathcal{O}_ X(C_ i)|_{C_ j})

for all 1 \leq i, j \leq n. If i = j there is nothing to prove. If i \not= j, then the canonical section 1 of \mathcal{O}_ X(C_ j) restricts to a nonzero (hence regular) section of \mathcal{O}_ X(C_ j)|_{C_ i} whose zero scheme is exactly C_ i \cap C_ j (scheme theoretic intersection). In other words, C_ i \cap C_ j is an effective Cartier divisor on C_ i and

\deg (\mathcal{O}_ X(C_ j)|_{C_ i}) = \deg (C_ i \cap C_ j)

by Varieties, Lemma 33.44.9. By symmetry we obtain the same (!) formula for the other side and the proof is complete. \square

In Situation 55.9.3 it is often convenient to think of \mathbf{Z}^{\oplus n} as the free abelian group on the set \{ C_1, \ldots , C_ n\} . We will indicate an element of this group as \sum a_ i C_ i; here we think of this as a formal sum although equivalently we may (and we sometimes do) think of such a sum as a Weil divisor on X supported on the special fibre X_ k. Now Lemma 55.9.6 allows us to define a symmetric bilinear form (\ \cdot \ ) on this free abelian group by the rule

55.9.6.1
\begin{equation} \label{models-equation-form} \left(\sum a_ i C_ i \cdot \sum b_ j C_ j\right) = \left\langle a, \mathcal{O}_ X(\sum b_ j C_ j) \right\rangle = \left\langle b, \mathcal{O}_ X(\sum a_ i C_ i) \right\rangle \end{equation}

We will prove some properties of this bilinear form.

Lemma 55.9.7. In Situation 55.9.3 the symmetric bilinear form (55.9.6.1) has the following properties

  1. (C_ i \cdot C_ j) \geq 0 if i \not= j with equality if and only if C_ i \cap C_ j = \emptyset ,

  2. (\sum m_ i C_ i \cdot C_ j) = 0,

  3. there is no nonempty proper subset I \subset \{ 1, \ldots , n\} such that (C_ i \cdot C_ j) = 0 for i \in I, j \not\in I.

  4. (\sum a_ i C_ i \cdot \sum a_ i C_ i) \leq 0 with equality if and only if there exists a q \in \mathbf{Q} such that a_ i = qm_ i for i = 1, \ldots , n,

Proof. In the proof of Lemma 55.9.6 we saw that (C_ i \cdot C_ j) = \deg (C_ i \cap C_ j) if i \not= j. This is \geq 0 and > 0 if and only if C_ i \cap C_ j \not= \emptyset . This proves (1).

Proof of (2). This is true because by Lemma 55.9.1 the invertible sheaf associated to \sum m_ i C_ i is trivial and the trivial sheaf has degree zero.

Proof of (3). This is expressing the fact that X_ k is connected (Lemma 55.9.4) via the description of the intersection products given in the proof of (1).

Part (4) follows from (1), (2), and (3) by Lemma 55.2.3. \square

Lemma 55.9.8. In Situation 55.9.3 set d = \gcd (m_1, \ldots , m_ n) and let D = \sum (m_ i/d)C_ i as an effective Cartier divisor. Then \mathcal{O}_ X(D) has order dividing d in \mathop{\mathrm{Pic}}\nolimits (X) and \mathcal{C}_{D/X} an invertible \mathcal{O}_ D-module of order dividing d in \mathop{\mathrm{Pic}}\nolimits (D).

Proof. We have

\mathcal{O}_ X(D)^{\otimes d} = \mathcal{O}_ X(dD) = \mathcal{O}_ X(X_ k) = \mathcal{O}_ X

by Lemma 55.9.1. We conclude as \mathcal{C}_{D/X} is the pullback of \mathcal{O}_ X(-D). \square

Lemma 55.9.9.reference In Situation 55.9.3 let d = \gcd (m_1, \ldots , m_ n). Let D = \sum (m_ i/d) C_ i as an effective Cartier divisor. Then there exists a sequence of effective Cartier divisors

(X_ k)_{red} = Z_0 \subset Z_1 \subset \ldots \subset Z_ m = D

such that Z_ j = Z_{j - 1} + C_{i_ j} for some i_ j \in \{ 1, \ldots , n\} for j = 1, \ldots , m and such that H^0(Z_ j, \mathcal{O}_{Z_ j}) is a field finite over k for j = 0, \ldots m.

Proof. The reduction D_{red} = (X_ k)_{red} = \sum C_ i is connected (Lemma 55.9.4) and proper over k. Hence H^0(D_{red}, \mathcal{O}) is a field and a finite extension of k by Varieties, Lemma 33.9.3. Thus the result for Z_0 = D_{red} = (X_ k)_{red} is true. Suppose that we have already constructed

(X_ k)_{red} = Z_0 \subset Z_1 \subset \ldots \subset Z_ t \subset D

with Z_ j = Z_{j - 1} + C_{i_ j} for some i_ j \in \{ 1, \ldots , n\} for j = 1, \ldots , t and such that H^0(Z_ j, \mathcal{O}_{Z_ j}) is a field finite over k for j = 0, \ldots , t. Write Z_ t = \sum a_ i C_ i with 1 \leq a_ i \leq m_ i/d. If a_ i = m_ i/d for all i, then Z_ t = D and the lemma is proved. If not, then a_ i < m_ i/d for some i and it follows that (Z_ t \cdot Z_ t) < 0 by Lemma 55.9.7. This means that (D - Z_ t \cdot Z_ t) > 0 because (D \cdot Z_ t) = 0 by the lemma. Thus we can find an i with a_ i < m_ i/d such that (C_ i \cdot Z_ t) > 0. Set Z_{t + 1} = Z_ t + C_ i and i_{t + 1} = i. Consider the short exact sequence

0 \to \mathcal{O}_ X(-Z_ t)|_{C_ i} \to \mathcal{O}_{Z_{t + 1}} \to \mathcal{O}_{Z_ t} \to 0

of Divisors, Lemma 31.14.3. By our choice of i we see that \mathcal{O}_ X(-Z_ t)|_{C_ i} is an invertible sheaf of negative degree on the proper curve C_ i, hence it has no nonzero global sections (Varieties, Lemma 33.44.12). We conclude that H^0(\mathcal{O}_{Z_{t + 1}}) \subset H^0(\mathcal{O}_{Z_ t}) is a field (this is clear but also follows from Algebra, Lemma 10.36.18) and a finite extension of k. Thus we have extended the sequence. Since the process must stop, for example because t \leq \sum (m_ i/d - 1), this finishes the proof. \square

Lemma 55.9.10.reference In Situation 55.9.3 let d = \gcd (m_1, \ldots , m_ n). Let D = \sum (m_ i/d) C_ i as an effective Cartier divisor on X. Then

1 - g_ C = d [\kappa : k] (1 - g_ D)

where g_ C is the genus of C, g_ D is the genus of D, and \kappa = H^0(D, \mathcal{O}_ D).

Proof. By Lemma 55.9.9 we see that \kappa is a field and a finite extension of k. Since also H^0(C, \mathcal{O}_ C) = K we see that the genus of C and D are defined (see Algebraic Curves, Definition 53.8.1) and we have g_ C = \dim _ K H^1(C, \mathcal{O}_ C) and g_ D = \dim _\kappa H^1(D, \mathcal{O}_ D). By Derived Categories of Schemes, Lemma 36.32.2 we have

1 - g_ C = \chi (C, \mathcal{O}_ C) = \chi (X_ k, \mathcal{O}_{X_ k}) = \dim _ k H^0(X_ k, \mathcal{O}_{X_ k}) - \dim _ k H^1(X_ k, \mathcal{O}_{X_ k})

We claim that

\chi (X_ k, \mathcal{O}_{X_ k}) = d \chi (D, \mathcal{O}_ D)

This will prove the lemma because

\chi (D, \mathcal{O}_ D) = \dim _ k H^0(D, \mathcal{O}_ D) - \dim _ k H^1(D, \mathcal{O}_ D) = [\kappa : k](1 - g_ D)

Observe that X_ k = dD as an effective Cartier divisor. To prove the claim we prove by induction on 1 \leq r \leq d that \chi (rD, \mathcal{O}_{rD}) = r \chi (D, \mathcal{O}_ D). The base case r = 1 is trivial. If 1 \leq r < d, then we consider the short exact sequence

0 \to \mathcal{O}_ X(rD)|_ D \to \mathcal{O}_{(r + 1)D} \to \mathcal{O}_{rD} \to 0

of Divisors, Lemma 31.14.3. By additivity of Euler characteristics (Varieties, Lemma 33.33.2) it suffices to prove that \chi (D, \mathcal{O}_ X(rD)|_ D) = \chi (D, \mathcal{O}_ D). This is true because \mathcal{O}_ X(rD)|_ D is a torsion element of \mathop{\mathrm{Pic}}\nolimits (D) (Lemma 55.9.8) and because the degree of a line bundle is additive (Varieties, Lemma 33.44.7) hence zero for torsion invertible sheaves. \square

Lemma 55.9.11. In Situation 55.9.3 given a pair of indices i, j such that C_ i and C_ j are exceptional curves of the first kind and C_ i \cap C_ j \not= \emptyset , then n = 2, m_1 = m_2 = 1, C_1 \cong \mathbf{P}^1_ k, C_2 \cong \mathbf{P}^1_ k, C_1 and C_2 meet in a k-rational point, and C has genus 0.

Proof. Choose isomorphisms C_ i = \mathbf{P}^1_{\kappa _ i} and C_ j = \mathbf{P}^1_{\kappa _ j}. The scheme C_ i \cap C_ j is a nonempty effective Cartier divisor in both C_ i and C_ j. Hence

(C_ i \cdot C_ j) = \deg (C_ i \cap C_ j) \geq \max ([\kappa _ i: k], [\kappa _ j : k])

The first equality was shown in the proof of Lemma 55.9.6. On the other hand, the self intersection (C_ i \cdot C_ i) is equal to the degree of \mathcal{O}_ X(C_ i) on C_ i which is -[\kappa _ i : k] as C_ i is an exceptional curve of the first kind. Similarly for C_ j. By Lemma 55.9.7

0 \geq (C_ i + C_ j)^2 = -[\kappa _ i : k] + 2(C_ i \cdot C_ j) - [\kappa _ j : k]

This implies that [\kappa _ i : k] = \deg (C_ i \cap C_ j) = [\kappa _ j : k] and that we have (C_ i + C_ j)^2 = 0. Looking at the lemma again we conclude that n = 2, \{ 1, 2\} = \{ i, j\} , and m_1 = m_2. Moreover, the scheme theoretic intersection C_ i \cap C_ j consists of a single point p with residue field \kappa and \kappa _ i \to \kappa \leftarrow \kappa _ j are isomorphisms. Let D = C_1 + C_2 as effective Cartier divisor on X. Observe that D is the scheme theoretic union of C_1 and C_2 (Divisors, Lemma 31.13.10) hence we have a short exact sequence

0 \to \mathcal{O}_ D \to \mathcal{O}_{C_1} \oplus \mathcal{O}_{C_2} \to \mathcal{O}_ p \to 0

by Morphisms, Lemma 29.4.6. Since we know the cohomology of C_ i \cong \mathbf{P}^1_\kappa (Cohomology of Schemes, Lemma 30.8.1) we conclude from the long exact cohomology sequence that H^0(D, \mathcal{O}_ D) = \kappa and H^1(D, \mathcal{O}_ D) = 0. By Lemma 55.9.10 we conclude

1 - g_ C = d[\kappa : k](1 - 0)

where d = m_1 = m_2. It follows that g_ C = 0 and d = m_1 = m_2 = 1 and \kappa = k. \square

[1] Observe that it may happen that the field \kappa _ i = H^0(C_ i, \mathcal{O}_{C_ i}) is strictly bigger than k. In this case every invertible module on C_ i has degree (as defined above) divisible by [\kappa _ i : k].

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