Exceptional curves of the first kind and their contractions

Assume the Axiom of Choice where it is inherited from the degree and intersection suppliers below (The Axiom of Choice). Let $X$ be a Noetherian scheme.

(a) Exceptional curves of the first kind. A closed subscheme $E\subseteq X$ (Closed immersions of schemes) is an exceptional curve of the first kind if:

1. $E$ is an effective Cartier divisor on $X$ (Effective cartier divisor, Cartier divisor);
2. there is a field $\kappa$ and an isomorphism ${\mathbb{P}}_{\kappa }^1\to E$ of schemes;
3. the normal sheaf ${\mathcal{N}}_{E\mathrm{/}X}={\mathcal{O}}_X(E){\mathrm{\mid }}_E$ pulls back to ${\mathcal{O}}_{{\mathbb{P}}_{\kappa }^1}(-1)$. Equivalently, its dual, the conormal sheaf ${\mathcal{O}}_X(-E){\mathrm{\mid }}_E$, pulls back to ${\mathcal{O}}_{{\mathbb{P}}_{\kappa }^1}(1)$ (Invertible sheaf of cartier divisor, Invertible sheaves).

Condition (3) is independent of the choice of the isomorphism in (2): any two such isomorphisms differ by an automorphism $\sigma$ of ${\mathbb{P}}_{\kappa }^1$, precomposition with $\sigma$ changes the pulled-back normal sheaf by ${\sigma }^{\ast }$, and ${\sigma }^{\ast }\mathcal{O}(-1)\cong \mathcal{O}(-1)$ because pullback induces an automorphism of the Picard group $\mathbb{Z}$, so it sends $\mathcal{O}(1)$ to $\mathcal{O}(1)$ or $\mathcal{O}(-1)$; the latter is excluded by preservation of $H^0$, since these have dimensions two and zero, respectively (Cohomology of O(d) on projective space). Thus it preserves $\mathcal{O}(-1)$. The twisting sheaves are classified by their twist index (The Picard group of the projective line, The twist index on the projective line is an isomorphism invariant). The degree of the normal sheaf is the integer $d$ with ${\mathcal{N}}_{E\mathrm{/}X}\cong {\mathcal{O}}_{{\mathbb{P}}_{\kappa }^1}(d)$.

(b) Contractions. Let $E\subseteq X$ be an exceptional curve of the first kind. A contraction of $E$ is a proper morphism $b\text{\hspace{0.17em}⁣}:X\to X^{\mathrm{\prime }}$ (Proper morphisms) such that:

1. $X^{\mathrm{\prime }}$ is a Noetherian scheme;
2. there is a closed point $x^{\mathrm{\prime }}\in X^{\mathrm{\prime }}$ whose local ring ${\mathcal{O}}_{X^{\mathrm{\prime }},x^{\mathrm{\prime }}}$ is regular of dimension $2$;
3. $X$ together with $b$ is the blowup of $X^{\mathrm{\prime }}$ at $x^{\mathrm{\prime }}$ in the sense of Blowup of a scheme along an ideal sheaf, and $E$ is identified with the scheme-theoretic exceptional fibre $b^{-1}(x^{\mathrm{\prime }})$.

Thus a contraction is exactly, up to unique isomorphism over $X$, the inverse of the blowing up of a regular point of a surface. The contracted curve is $E$. Uniqueness with an initially unspecified target is a separate theorem proved later on this page.

(c) Contraction notation. If $f\text{\hspace{0.17em}⁣}:X\to Y$ is a morphism of schemes and $E\subseteq X$ is an integral curve (Integral schemes), we say that $E$ is contracted by $f$ if $f(E)$ is a single point of $Y$. This is a condition on $f$ alone and does not presuppose that $f$ is a blowup or that $E$ is exceptional.

(d) Dictionary on a regular projective surface. Let $k$ be a field, let $X$ be an integral regular projective surface over $k$, and let $E$ be an integral effective Cartier divisor on $X$ (Intersection numbers of Cartier divisors on a smooth projective surface). Suppose $E$ is $k$-isomorphic to ${\mathbb{P}}_{\kappa }^1$ for a finite extension $\kappa \mathrm{/}k$, where $\kappa =H^0(E,{\mathcal{O}}_E)$ is the constant field of $E$, not the function field $\kappa (E)$. Then $E$ is exceptional of the first kind exactly when

$E\cdot E=-[\kappa :k]\mathrm{.}$

In particular, for $E\cong {\mathbb{P}}_k^1$ over $k$ the criterion is $E\cdot E=-1$. Indeed the normal line bundle is $\mathcal{O}(d)$ for a unique $d\in \mathbb{Z}$ by The Picard group of the projective line and The twist index on the projective line is an isomorphism invariant, and the projective-space cohomology calculation (Cohomology of O(d) on projective space) gives

${\chi }_k({\mathbb{P}}_{\kappa }^1,\mathcal{O}(d))=[\kappa :k](d+1),$

because restriction of scalars along $\kappa \mathrm{/}k$ multiplies the dimensions of the $k$-vector spaces $H^q({\mathbb{P}}_{\kappa }^1,\mathcal{O}(d))$ by $[\kappa :k]$ and $H^0=\kappa [x_0,x_1{]}_d$, $H^1=\kappa [x_0,x_1{]}_{-d-2}$ have dimensions $d+1$ and $-d-1$ for $d\ge 0$ and $d\le -2$ respectively, with all other terms zero. By Degree of an invertible sheaf on a proper one-dimensional scheme the $k$-degree of the invertible sheaf ${\mathcal{O}}_{{\mathbb{P}}_{\kappa }^1}(d)$ is therefore $[\kappa :k]d$, and the restriction-degree identity (Intersection with a curve is the degree of the restriction) applied to the effective Cartier divisors $E$ and $E$ on $X$ gives

$E\cdot E={\mathrm{deg}}_E\left({\mathcal{N}}_{E\mathrm{/}X}\right)=[\kappa :k]d\mathrm{.}$

Hence $E$ is exceptional of the first kind if and only if the normal twist is $d=-1$, which is equivalent to $E\cdot E=-[\kappa :k]$, and to $E\cdot E=-1$ when $\kappa =k$.

• The dictionary (d) is the reason the notion of exceptional curve of the first kind can be checked in practice on a regular projective surface: the normal twist $-1$ is an intersection number, and intersection numbers are computable without constructing a contraction. It says nothing about existence of a contraction of a given exceptional curve; that question is taken up by the contraction and factorisation theorems below.
• The scheme-theoretic exceptional fibre $b^{-1}(x^{\mathrm{\prime }})$ in (b) is the fibre product $X{\times }_{X^{\mathrm{\prime }}}\mathrm{Spec}\kappa (x^{\mathrm{\prime }})$; it is a $\kappa (x^{\mathrm{\prime }})$-scheme isomorphic to ${\mathbb{P}}_{\kappa (x^{\mathrm{\prime }})}^1$ (Blowup of a scheme along an ideal sheaf). Condition (3) records the extra requirement that the given $E$ is that fibre, not merely a curve lying over $x^{\mathrm{\prime }}$.
• The intersection number $E\cdot E$ in (d) is defined on the regular projective surface in (d), over an arbitrary field; no embedding in projective space and no point of $X$ outside $E$ is used (Intersection numbers of Cartier divisors on a smooth projective surface).
• The Axiom of Choice is inherited: the degree of an invertible sheaf on a proper curve is defined through Euler characteristics, and the intersection product through the degree (Degree of an invertible sheaf on a proper one-dimensional scheme). No other choice principle is used, and no choice is made in the definition itself.