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Dualizing the determinant identity in the Euler sequence on complex projective space giveswhere $H$ is a hyperplane divisor.

For a Noetherian integral scheme $X$ that is regular in codimension one,where $H=X\times {\mathbb{P}}^{n-1}$ is a hyperplane divisor. Restriction to $X\times {\mathbb{A}}^n$ gives the first summand, while restriction to the generic projective-space fiber detects the coefficient of $H$.

The two pairs of regular functionsdefine maps to the projective line on the loci where their respective coordinates do not vanish simultaneously. Those loci cover $Y$, since simultaneous failure would force $x=y=z=w=0$. On their overlap the equation $xy=zw$ says that the two projective points are equal. They therefore glue to a morphism

Let $[A:B]$ be the homogeneous coordinates on ${\mathbb{P}}_k^1$. The pullback of the hyperplane divisor $V(A)$ has local equation $x$ on the first chart and $z$ on the second. It is therefore exactly $D_Y=V(x,z)$. Compatibility of the pullback of a sheaf of modules with the line bundle associated to a divisor givesThis is the ruling morphism of the punctured three-dimensional affine quadric cone associated with $D_Y$.

The scheme ${\mathbb{P}}_k^3$ is Noetherian, integral, separated, and regular. Every open subscheme inherits these properties, so $U={\mathbb{P}}_k^3\setminus C$ satisfies $(\star )$. Because $C$ has dimension one, it has codimension two in ${\mathbb{P}}_k^3$ and contains no prime Weil divisor. The localization sequence for the divisor class group therefore makes restriction an isomorphismThe hyperplane divisor generates the class group of projective space, so

A Cartier divisor on an integral scheme is an open cover $(U_i)$ together with nonzero rational functions $f_i$ such that every ratio $f_i/f_j$ is a regular unit on $U_i\cap U_j$.

Every prime Weil divisor on ${\mathbb{P}}_k^n$ is cut out by an irreducible homogeneous polynomial because the polynomial ring $k[x_0,\dots ,x_n]$ is a unique factorization domain. Hence any Weil divisor can be represented by a homogeneous rational expressionof some total degree $d$. On the standard chart $U_i=D_{+}(x_i)$, put $f_i=F/x_i^d$. This is a degree-zero rational function, and on $U_i\cap U_j$,is a regular unit. These local equations form a Cartier divisor whose associated Weil divisor is the original one.

Now put $P=X\times {\mathbb{P}}^n$ and let $H=X\times {\mathbb{P}}^{n-1}$ be the hyperplane divisor at infinity. Its complement is $X\times {\mathbb{A}}^n$. Iterating the given invariance under multiplication by ${\mathbb{A}}^1$ givesThe localization sequence for the divisor class group shows that every class on $P$ is a pullback of a class on $X$ plus an integer multiple of $[H]$. Restriction to the generic fiber ${\mathbb{P}}_{K(X)}^n$ kills pullbacks from $X$ and sends $[H]$ to the generator of $\mathrm{Cl}({\mathbb{P}}_{K(X)}^n)\cong \mathbb{Z}$. Therefore the sum is direct, provingThis is the divisor class group of a projective-space bundle with trivial vector bundle.

An irreducible complex analytic hypersurface in a complex manifold $X$ is a closed irreducible analytic subset of pure complex codimension one. A local defining function of a complex analytic hypersurface $Y$ at $x$ is a holomorphic function $f$ on a neighbourhood $U$ such thatThe necessary local algebra is that the stalk ${\mathcal{O}}_{X,x}$ is a regular local ring, hence a unique factorization domain, and that the local branches of a hypersurface germ determine finitely many height-one prime ideals. Each is principal; the product of their generators gives $f$, and removing repeated factors makes it reduced. This also covers a globally irreducible hypersurface that has several local branches at a singular point.

A divisor on a complex manifold is a locally finite formal sum $D={\sum }_Y{n}_{Y}Y$ of irreducible analytic hypersurfaces with integer coefficients. On a sufficiently small $U_{\alpha }$, local defining functions give a meromorphic equation $f_{\alpha }$ for $D$. The quotients $f_{\alpha }/f_{\beta }$ are nowhere-zero holomorphic functions. Gluing frames byproduces the holomorphic line bundle associated to a divisor $[D]$, and $f_{\alpha }{e}_{\alpha }$ gives its canonical meromorphic section with divisor $D$.

The Euler sequence on complex projective spaceimplies $\det T{\mathbb{CP}}^n\cong \mathcal{O}(n+1)$. Taking the dual determinant yields the canonical bundle of complex projective spacewhere $H$ is a hyperplane divisor.

The hypotheses on the homogeneous polynomial $p$ say thatis a smooth projective hypersurface of degree $k$, so its divisor line bundle is $[V]\cong \mathcal{O}(kH)$. The Adjunction formula givesThis is the canonical bundle of a smooth projective hypersurface.

Now fix an isomorphism $\Phi :[P]\to [Q]$ and regard $s_P$ and ${\Phi }^{-1}{s}_Q$ as holomorphic sections of the same holomorphic line bundle. They have no common zero because $P\ne Q$. Their homogeneous coordinates therefore define a well-defined holomorphic mapIn a local frame, the quotientis a meromorphic function with divisor $(f)=P-Q$. Thus $F^{-1}(0)=P$ and $F^{-1}(\infty )=Q$, both with multiplicity one. The degree of a holomorphic map $F$ is therefore one. A nonconstant degree-one holomorphic map between compact connected Riemann surfaces is a biholomorphism, so the displayed map is biholomorphic.

A projective hyperplane in ${\mathbb{CP}}^n$ is the zero set of a nonzero complex-linear functional on ${\mathbb{C}}^{n+1}$. It is isomorphic to ${\mathbb{CP}}^{n-1}$ and determines a hyperplane divisor.