Let \(\pi: \tilde{X} \rightarrow X\) be a monoidal transformation, and let \(D\) be a very ample divisor on \(X\). Show that \(2 \pi^* D-E\) is ample on \(\tilde{X}\).
Hint: Use a suitable generalization of (I, Ex. 7.5) to curves in \(\PP^n\).
Solution
Let \[ P\in X \] be the centre of the monoidal transformation, and let \[ E\subseteq\widetilde X \] be the exceptional curve. Put \[ A=2\pi^*D-E. \]
1. If \(C\subseteq X\) is an irreducible curve and\[ r=\mu_P(C), \]where \(r=0\) when \(P\notin C\), then\[ \boxed{r\le D\cdot C.} \]
Because \(D\) is very ample, its complete linear system embeds \[ \iota_D:X\hookrightarrow\PP^N \] and \[ D\cdot C=\deg\iota_D(C). \] The multiplicity of \(C\) at \(P\) is unchanged by this closed immersion.
Choose a hyperplane \(H\subseteq\PP^N\) through \(\iota_D(P)\) which does not contain \(\iota_D(C)\) and is general among such hyperplanes. The local intersection multiplicity at \(P\) satisfies \[ i_P(C,H)\ge\mu_P(C)=r, \] while the sum of all local intersection multiplicities with \(H\) is \[ H\cdot\iota_D(C)=\deg\iota_D(C)=D\cdot C. \] Therefore \[ r\le D\cdot C. \] This is the required projective-space generalization of the multiplicity bound in P-AGH75MULTBOUND.
2. The divisor \(A\) has positive intersection with the exceptional curve:\[ \boxed{A\cdot E=1.} \]
The blowup intersection formulas FE-SRFBLOW give \[ \pi^*D\cdot E=0, \qquad E^2=-1. \] Hence \[ A\cdot E =(2\pi^*D-E)\cdot E =-E^2 =1. \]
3. Let \(\Gamma\subseteq\widetilde X\) be an irreducible curve with\[ \Gamma\ne E. \]Then \(\Gamma\) is the strict transform of an irreducible curve\(C\subseteq X\).
The blowup map \[ \pi:\widetilde X\longrightarrow X \] is an isomorphism over \(X\setminus\{P\}\). Since \(\Gamma\) is not contained in the exceptional locus \(E\), its image is an irreducible curve \(C\subseteq X\), and \(\Gamma\) is the closure of the inverse image of \(C\setminus\{P\}\). This is exactly the strict transform of \(C\).
4. For every irreducible curve \(\Gamma\ne E\) on \(\widetilde X\),\[ \boxed{A\cdot\Gamma>0.} \]
Let \(C=\pi(\Gamma)\) and put \[ r=\mu_P(C), \] again taking \(r=0\) if \(P\notin C\). The strict-transform formula gives \[ \Gamma\equiv\pi^*C-rE. \] Using the blowup intersection identities, \[ \begin{aligned} A\cdot\Gamma &=(2\pi^*D-E)\cdot(\pi^*C-rE)\\ &=2D\cdot C-r. \end{aligned} \] By step 1, \[ r\le D\cdot C, \] so \[ A\cdot\Gamma \ge D\cdot C. \] Since \(D\) is very ample, its restriction to the irreducible curve \(C\) has positive degree. Thus \[ D\cdot C>0, \] and consequently \(A\cdot\Gamma>0\).
5. The self-intersection of \(A\) is positive:\[ \boxed{A^2=4D^2-1>0.} \]
Again using \[ \pi^*D\cdot E=0, \qquad E^2=-1, \] one obtains \[ \begin{aligned} A^2 &=(2\pi^*D-E)^2\\ &=4(\pi^*D)^2+E^2\\ &=4D^2-1. \end{aligned} \] Because \(D\) is very ample on the projective surface \(X\), it gives an embedding of \(X\) and \[ D^2=\deg_D(X) \] is a positive integer. Hence \[ 4D^2-1\ge3>0. \]
6. The divisor\[ \boxed{2\pi^*D-E} \]is ample on \(\widetilde X\).
Step 2 gives positive intersection with the exceptional curve. Step 4 gives positive intersection with every other irreducible curve on \(\widetilde X\), and step 5 gives positive self-intersection. Therefore the Nakai–Moishezon criterion T-SRFNAKAI applies and shows that \[ 2\pi^*D-E \] is ample.
7. Q.E.D.
Steps <1>1–<1>5 verify the numerical hypotheses of Nakai–Moishezon, and step 6 gives the required ampleness.