The Stacks project

Proof. The second equality follows from the definition of the bilinear form $(\ \cdot \ )$ in (55.9.6.1) and Lemma 55.9.6. To see the first equality we distinguish two cases. Namely, if $C_ i \not\subset D$, then $D'$ is the scheme theoretic union of $D$ and $C_ i$ (by Divisors, Lemma 31.13.10) and we get a short exact sequence

by Morphisms, Lemma 29.4.6. Since we also have an exact sequence

(Divisors, Remark 31.14.11) we conclude that the claim holds by additivity of euler characteristics (Varieties, Lemma 33.33.2). On the other hand, if $C_ i \subset D$ then we get an exact sequence

by Divisors, Lemma 31.14.3 and we immediately see the lemma holds. $\square$

Proof. Our basic tool will be Derived Categories of Schemes, Lemma 36.32.2 which shows that

Choose a sequence of effective Cartier divisors

such that $D_{j + 1} = D_ j + C_{i_ j}$ for each $j$. (It is clear that we can choose such a sequence by decreasing one nonzero multiplicity of $D_{j + 1}$ one step at a time.) Applying Lemma 55.11.1 starting with $\chi (\mathcal{O}_{D_0}) = 0$ we get

Perhaps the last equality deserves some explanation. Namely, since $\sum _ j C_{i_ j} = \sum m_ i C_ i$ we have $(\sum _ j C_{i_ j} \cdot \sum _ j C_{i_ j}) = 0$ by Lemma 55.9.7. Thus we see that

by splitting this product into “nondiagonal” and “diagonal” terms. Note that $\kappa _ i$ is a field finite over $k$ by Varieties, Lemma 33.26.2. Hence the genus of $C_ i$ is defined and we have $\chi (C_ i, \mathcal{O}_{C_ i}) = [\kappa _ i : k](1 - g_ i)$. Putting everything together and rearranging terms we get

which is what the lemma says too. $\square$

Proof. (In the proof of Lemma 55.11.2 we have seen that the quantities used in the statement of the lemma are well defined.) We have to verify the conditions (1) – (5) of Definition 55.3.1.

Condition (1) is immediate.

Condition (2). Symmetry of the matrix $(C_ i \cdot C_ j)$ follows from Equation (55.9.6.1) and Lemma 55.9.6. Nonnegativity of $(C_ i \cdot C_ j)$ for $i \not= j$ is part (1) of Lemma 55.9.7.

Condition (3) is part (3) of Lemma 55.9.7.

Condition (4) is part (2) of Lemma 55.9.7.

Condition (5) follows from the fact that $(C_ i \cdot C_ j)$ is the degree of an invertible module on $C_ i$ which is divisible by $[\kappa _ i : k]$, see Varieties, Lemma 33.44.10.

The genus formula proved in Lemma 55.11.2 tells us that the numerical type has the genus as stated, see Definition 55.3.4. $\square$

Proof. If the numerical type is minimal, then there is no $i$ with $g_ i = 0$ and $(C_ i \cdot C_ i) = -[\kappa _ i: k]$, see Definition 55.3.12. Certainly, this implies that none of the curves $C_ i$ are exceptional curves of the first kind.

Conversely, suppose that the numerical type is not minimal. Then there exists an $i$ such that $g_ i = 0$ and $(C_ i \cdot C_ i) = -[\kappa _ i: k]$. We claim this implies that $C_ i$ is an exceptional curve of the first kind. Namely, the invertible sheaf $\mathcal{O}_ X(-C_ i)|_{C_ i}$ has degree $-(C_ i \cdot C_ i) = [\kappa _ i : k]$ when $C_ i$ is viewed as a proper curve over $k$, hence has degree $1$ when $C_ i$ is viewed as a proper curve over $\kappa _ i$. Applying Algebraic Curves, Proposition 53.10.4 we conclude that $C_ i \cong \mathbf{P}^1_{\kappa _ i}$ as schemes over $\kappa _ i$. Since the Picard group of $\mathbf{P}^1$ over a field is $\mathbf{Z}$, we see that the normal sheaf of $C_ i$ in $X$ is isomorphic to $\mathcal{O}_{\mathbf{P}_{\kappa _ i}}(-1)$ and the proof is complete. $\square$

Proof. Since $X \to \mathop{\mathrm{Spec}}(R)$ is proper, the $K$-rational point extends to a morphism $a : \mathop{\mathrm{Spec}}(R) \to X$ by the valuative criterion of properness (Morphisms, Lemma 29.42.1). Let $x \in X$ be the image under $a$ of the closed point of $\mathop{\mathrm{Spec}}(R)$. Then $a$ corresponds to an $R$-algebra homomorphism $\psi : \mathcal{O}_{X, x} \to R$ (see Schemes, Section 26.13). It follows that $\pi \not\in \mathfrak m_ x^2$ (since the image of $\pi $ in $R$ is not in $\mathfrak m_ R^2$). Hence $\mathcal{O}_{X_ k, x} = \mathcal{O}_{X, x}/\pi \mathcal{O}_{X, x}$ is regular (Algebra, Lemma 10.106.3). Then $X_ k \to \mathop{\mathrm{Spec}}(k)$ is smooth at $x$ by Algebra, Lemma 10.140.5. It follows that $x$ is contained in a unique irreducible component $C_ i$ of $X_ k$, that $\mathcal{O}_{C_ i, x} = \mathcal{O}_{X_ k, x}$, and that $m_ i = 1$. The fact that $C_ i$ has a $k$-rational point implies that the field $\kappa _ i = H^0(C_ i, \mathcal{O}_{C_ i})$ (Varieties, Lemma 33.26.2) is equal to $k$. This proves (1). We have $H^0(X_ k, \mathcal{O}_{X_ k}) = k$ because $H^0(X_ k, \mathcal{O}_{X_ k})$ is a field extension of $k$ (Lemma 55.9.9) which maps to $H^0(C_ i, \mathcal{O}_{C_ i}) = k$. The genus equality follows from Lemma 55.9.10. $\square$

Proof. By vanishing of cohomology in degrees $\geq 2$ over $X_ k$ (Cohomology, Proposition 20.20.7) any surjection of abelian sheaves on $X_ k$ induces a surjection on $H^1$. Consider the sequence

of Lemma 55.9.9. Since the field maps $H^0(Z_ j, \mathcal{O}_{Z_ j}) \to H^0((X_ k)_{red}, \mathcal{O}_{(X_ k)_{red}}) = k$ are injective we conclude that $H^0(Z_ j, \mathcal{O}_{Z_ j}) = k$ for $j = 0, \ldots , m$. It follows that $H^0(X_ k, \mathcal{O}_{X_ k}) \to H^0(Z_{m - 1}, \mathcal{O}_{Z_{m - 1}})$ is surjective. Let $C = C_{i_ m}$. Then $X_ k = Z_{m - 1} + C$. Let $\mathcal{L} = \mathcal{O}_ X(-Z_{m - 1})|_ C$. Then $\mathcal{L}$ is an invertible $\mathcal{O}_ C$-module. As in the proof of Lemma 55.9.9 there is an exact sequence

of coherent sheaves on $X_ k$. We conclude that we get a short exact sequence

The degree of $\mathcal{L}$ on $C$ over $k$ is

Set $\kappa = H^0(C, \mathcal{O}_ C)$ and $w = [\kappa : k]$. By definition of the degree of an invertible sheaf we see that

where $g_ C$ is the genus of $C$. This expression is $< 0$ as $X$ is minimal and hence $C$ is not an exceptional curve of the first kind (see proof of Lemma 55.11.5). Thus $\dim _ k H^1(C, \mathcal{L}) > 0$ which finishes the proof. $\square$

Proof. We are going to prove the inequality

for all connected reduced effective Cartier divisors $D \subset (X_ k)_{red}$ containing $x$ by induction on the number of irreducible components of $D$. Here $g_{top}(D) = 1 - m + e$ where $m$ is the number of irreducible components of $D$ and $e$ is the number of unordered pairs of components of $D$ which meet.

Base case: $D$ has one irreducible component. Then $D = C_ i$ is the unique irreducible component containing $x$. In this case $\dim _ k H^1(D, \mathcal{O}_ D) = g_ i$ and $g_{top}(D) = 0$. Since $C_ i$ has a $k$-rational smooth point it is geometrically integral (Varieties, Lemma 33.25.10). It follows that $g_ i$ is the genus of $C_{i, \overline{k}}$ (Algebraic Curves, Lemma 53.8.2). It also follows that $g_{geom}(D/k)$ is the genus of the normalization $C_{i, \overline{k}}^\nu $ of $C_{i, \overline{k}}$. Applying Algebraic Curves, Lemma 53.18.4 to the normalization morphism $C_{i, \overline{k}}^\nu \to C_{i, \overline{k}}$ we get

Combining the above we conclude that $\dim _ k H^1(D, \mathcal{O}_ D) \geq g_{top}(D) + g_{geom}(D/k)$ in this case.

Induction step. Suppose we have $D$ with more than $1$ irreducible component. Then we can write $D = C_ i + D'$ where $x \in D'$ and $D'$ is still connected. This is an exercise in graph theory we leave to the reader (hint: let $C_ i$ be the component of $D$ which is farthest from $x$). We compute how the invariants change. As $x \in D'$ we have $H^0(D, \mathcal{O}_ D) = H^0(D', \mathcal{O}_{D'}) = k$. Looking at the short exact sequence of sheaves

(Morphisms, Lemma 29.4.6) and using additivity of euler characteristics we find

Here as in Lemma 55.11.3 we set $w_ i = [\kappa _ i : k]$, $\kappa _ i = H^0(C_ i, \mathcal{O}_{C_ i})$, $g_ i$ is the genus of $C_ i$, and $a_{ij} = (C_ i \cdot C_ j)$. We have

We have

by Algebraic Curves, Lemma 53.18.1. Combining these with our induction hypothesis, we conclude that it suffices to show that

is nonnegative. In fact, we have

The second inequality by Algebraic Curves, Lemma 53.18.5. On the other hand, since $w_ i$ divides $a_{ij}$ (Varieties, Lemma 33.44.10) it is clear that

because there is at least one $C_ j \subset D'$ which meets $C_ i$. $\square$

Proof. Looking over the proof of Lemma 55.11.9 we see that in order to get equality, the inequalities (55.11.9.1), (55.11.9.2), and (55.11.9.3) have to be equalities.

Let $C_ i$ be the irreducible component containing $x$. Equality in (55.11.9.1) shows via Algebraic Curves, Lemma 53.18.4 that $C_{i, \overline{k}}^\nu \to C_{i, \overline{k}}$ is an isomorphism. Hence $C_{i, \overline{k}}$ is smooth and part (1) holds.

Next, let $C_ i \subset X_ k$ be another irreducible component. Then we may assume we have $D = D' + C_ i$ as in the induction step in the proof of Lemma 55.11.9. Equality in (55.11.9.2) immediately implies that $\kappa _ i/k$ is finite separable. Equality in (55.11.9.3) implies either $a_{ij} = 1$ for some $j$ or that there is a unique $C_ j \subset D'$ meeting $C_ i$ and $a_{ij} = w_ i$. In both cases we find that $C_ i$ has a $\kappa _ i$-rational point $c$ and $c = C_ i \cap C_ j$ scheme theoretically. Since $\mathcal{O}_{X, c}$ is a regular local ring, this implies that the local equations of $C_ i$ and $C_ j$ form a regular system of parameters in the local ring $\mathcal{O}_{X, c}$. Then $\mathcal{O}_{C_ i, c}$ is regular by (Algebra, Lemma 10.106.3). We conclude that $C_ i \to \mathop{\mathrm{Spec}}(\kappa _ i)$ is smooth at $c$ (Algebra, Lemma 10.140.5). It follows that $C_ i$ is geometrically integral over $\kappa _ i$ (Varieties, Lemma 33.25.10). To finish we have to show that $C_ i$ is smooth over $\kappa _ i$. Observe that

where there are $[\kappa _ i : k]$-summands. Thus if $C_ i$ is not smooth over $\kappa _ i$, then each of these curves is not smooth, then these curves are not normal and the normalization morphism drops the genus (Algebraic Curves, Lemma 53.18.4) which is disallowed because it would drop the geometric genus of $C_ i/k$ contradicting $[\kappa _ i : k] g_ i = g_{geom}(C_ i/k)$. $\square$