As in (8.24), let \(X\) be a nonsingular integral variety over an algebraically closed field \(k\), let \(Y\) be a nonempty nonsingular integral closed subvariety of codimension \(r \geq 2\), let \(\pi: \tilde X \to X\) be the blowing-up of \(X\) along \(Y\), and let \(Y' = \inverseof{\pi}(Y)\).
Show that the maps \(\pi^*: \Pic X \to \Pic \tilde X\), and \(\ZZ \to \Pic \tilde X\) defined by \(n \mapsto\) the class of \(nY'\), give rise to an isomorphism \(\Pic \tilde X \cong \Pic X \oplus \ZZ\).
Show that \[ \omega_{\tilde X} \cong \pi^* \omega_X \tensor \mcl((r-1) Y') . \]
Hint
By (a), write \(\omega_{\tilde X}\cong\pi^*\mcm\tensor\mcl(qY')\). Restrict to \(\tilde X\setminus Y'\cong X\setminus Y\) to show that \(\mcm\cong\omega_X\). Adjunction should give \(\omega_{Y'}\cong(\pi|_{Y'})^*\omega_X\tensor\OO_{Y'}(-q-1)\). For a closed point \(y\in Y\) and its fibre \(Z\subseteq Y'\), show that \(\omega_Z\cong\OO_Z(-q-1)\). Since \(Z\cong\PP^{r-1}\) and \(\omega_Z\cong\OO_Z(-r)\), this determines \(q\).
Solution
Write \(E=Y'\), \(U=X\setminus Y\), and \(\widetilde U=\widetilde X\setminus E\). Let \(i:Y\hookrightarrow X\) and \(p:E\to Y\) be the natural maps. All divisor classes are identified with invertible sheaf classes by \(D\mapsto\OO(D)\) on the nonsingular varieties.
1. The varieties \(\widetilde X\) and \(E\) are nonsingular and integral, and\[ E\cong\PP_Y(\mathcal I_Y/\mathcal I_Y^2),\qquad \OO_{\widetilde X}(E)|_E\cong\OO_E(-1),\qquad \widetilde U\cong U. \]
The blowup theorem for a smooth center gives smoothness, the displayed exceptional projective bundle in the quotient convention, and \(\OO_{\widetilde X}(-E)|_E\cong\OO_E(1)\) [@Har10a, Proposition II.8.24]. Dualizing gives the asserted sign for \(\OO(E)|_E\). The conormal bundle \(\mathcal I_Y/\mathcal I_Y^2\) has rank \(r\). Since \(Y\) is integral, its projective bundle is integral: local trivializations are projective spaces over integral opens and overlap over the generic point of \(Y\). Thus \(E\) is an irreducible prime divisor. The blowup of the integral \(X\) along the proper center is integral, and is an isomorphism off that center, by the blowup construction. For every closed point \(y\in Y\), its fibre \(Z=p^{-1}(y)\) is \(\PP_k^{r-1}\) and \[ \OO_{\widetilde X}(E)|_Z\cong\OO_{\PP^{r-1}}(-1). \]
2. Restriction gives an isomorphism \(\Pic(X)\to\Pic(U)\), and the kernel of \(\Pic(\widetilde X)\to\Pic(\widetilde U)\) is generated by \([\OO_{\widetilde X}(E)]\).
On a regular integral noetherian variety, the Cartier--Weil comparison identifies \(\Pic\) with the Weil divisor class group [@Har10a, Corollary II.6.16]. Removing \(Y\), whose codimension is at least two, removes no prime divisor of \(X\). Every prime divisor on \(U\) closes to one on \(X\), and the common function field gives the same principal-divisor relations. Thus \(\Cl(X)\to\Cl(U)\) is an isomorphism, and so is the Picard restriction.
For \(\widetilde X\), the omitted set is the single prime divisor \(E\). The divisor restriction sequence [@Har10a, Proposition II.6.5] gives \[ \ZZ[E]\longrightarrow\Cl(\widetilde X) \longrightarrow\Cl(\widetilde U)\longrightarrow0. \] Applying the same Cartier–Weil identifications proves the kernel assertion.
3. The isomorphism in (a) is\[ \boxed{\Pic(X)\oplus\ZZ\xrightarrow{\cong}\Pic(\widetilde X),\qquad ([M],q)\longmapsto[\pi^*M\otimes\OO_{\widetilde X}(qE)].} \]
2. This proves surjectivity.
1. This gives \(\OO_Z(-q)\cong\OO_Z\).Since \(r-1\ge1\), \(\Pic(Z)=\ZZ\) with generator \(\OO_Z(1)\), so \(q=0\) [@Har10a, Proposition II.6.4].Thus the kernel is zero.
4. There is an integer \(q\) with\[ \omega_{\widetilde X}\cong\pi^*\omega_X\otimes\OO_{\widetilde X}(qE), \qquad \omega_E\cong(i\circ p)^*\omega_X\otimes\OO_E(-q-1). \]
Step 3 first writes \(\omega_{\widetilde X}=\pi^*M\otimes\OO(qE)\). On \(\widetilde U\cong U\), the differential sheaves and their top exterior powers agree under the isomorphism. Thus \(M|_U\cong\omega_X|_U\). Injectivity in step 2 implies \(M\cong\omega_X\) on \(X\).
1. This gives the second formula, including the exponent \(-q-1\).
5. Restricting to a fibre determines \(q=r-1\) and proves (b).
Take a closed point \(y\in Y\), put \(m=\dim Y\), and let \(Z=p^{-1}(y)\cong\PP_k^{r-1}\). A projective-bundle trivialization over a neighborhood of \(y\) identifies \(p\) with projection from a product. Consequently the conormal sheaf of this fibre is \[ \mathcal I_{Z/E}/\mathcal I_{Z/E}^2 \cong(\mathfrak m_y/\mathfrak m_y^2)\otimes_k\OO_Z \cong\OO_Z^{\oplus m}. \] This follows directly by pulling back the ideal of \(y\) through the flat product projection; the residue cotangent space has dimension \(m\) because \(Y\) is nonsingular. The exact conormal sequence for the nonsingular closed immersion \(Z\hookrightarrow E\), followed by determinants, therefore gives \[ \omega_E|_Z\cong\omega_Z, \] up to tensoring with the constant one-dimensional vector space \(\det(\mathfrak m_y/\mathfrak m_y^2)\), whose associated invertible sheaf is trivial.
On \(Z\), the pullback of \(\omega_X\) from step 4 is also a constant invertible sheaf, since the map to \(X\) factors through \(y\). Hence \(\omega_Z\cong\OO_Z(-q-1)\). But the canonical sheaf of \(\PP^{r-1}\) is \(\OO_Z(-r)\) [@Har10a, Example II.8.20.1]. Its Picard group is freely generated by \(\OO_Z(1)\), so \(-q-1=-r\) and \(q=r-1\). Substituting into step 4 gives \[ \boxed{\omega_{\widetilde X}\cong\pi^*\omega_X\otimes\OO_{\widetilde X}((r-1)E).} \]
6. Q.E.D.
Step 3 proves (a), and steps <1>4–<1>5 prove (b) through the stated adjunction and exceptional-fibre comparison.