Let \(X\) be a projective variety of dimension \(r\) in \(\PP^n\), with \(n \geq r + 2\). Show that for a suitable choice of a point \(P \notin X\) and a linear subspace \(\PP^{n-1} \subseteq \PP^n\), the projection from \(P\) to \(\PP^{n-1}\) induces a birational morphism of \(X\) onto its image \(X' \subseteq \PP^{n-1}\).
This shows in particular that the birational map making \(X\) birational to a hypersurface can be obtained by a finite number of such projections.
Solution
Let \(K=k(X)\) be the function field of \(X\). We use that \(k\) is algebraically closed, hence perfect, so every finitely generated extension of \(k\) is separably generated [@Har10a, Theorem I.4.8A].
1. After a projective linear change of coordinates, the rational functions\[ u_i=\frac{x_i}{x_0}\qquad(1\le i\le r+1) \]generate \(K\) over \(k\).
Choose a hyperplane not containing \(X\) and call its equation \(x_0=0\). On the dense affine open \(X\cap D_+(x_0)\), the coordinate ratios \(x_i/x_0\), \(1\le i\le n\), generate \(K\) as a field over \(k\).
Since \(K/k\) is separably generated of transcendence degree \(r\), Theorem I.4.7A permits, after an invertible linear change among these affine linear coordinates, a separating transcendence basis \(u_1,\ldots,u_r\) chosen from them. Thus \(K\) is finite separable over \[ F=k(u_1,\ldots,u_r). \] The remaining affine coordinates generate this finite extension. By the primitive element theorem [@Har10a, Theorem I.4.6A], a suitable \(k\)-linear combination of those remaining coordinates is a primitive element \(u_{r+1}\) for \(K/F\). Replacing one remaining projective coordinate by the same linear combination gives \[ K=k(u_1,\ldots,u_r,u_{r+1}). \]
The choices just made persist on a nonempty Zariski-open set of linear coordinate systems. For the transcendence basis, nonvanishing of \(du_1\wedge\cdots\wedge du_r\) in \(\Omega_{K/k}^r\) is an open nonvanishing condition on the linear coefficients. For the primitive element, if \(K/F\) is replaced by a finite normal extension containing it, failure of a linear combination to separate two distinct \(F\)-embeddings is one linear equation in its coefficients; only finitely many pairs of embeddings occur. Hence primitive linear combinations form a nonempty open subset of the relevant coefficient space. Finally, the condition that the last coordinate point \([0:\cdots:0:1]\) lie outside the transformed \(X\) is also open and nonempty, because \(X\subsetneq\PP^n\). The parameter space of projective coordinate systems is irreducible, so these finitely many nonempty open conditions can be imposed simultaneously. We henceforth use such a coordinate system.
2. Projection from\[ P=[0:\cdots:0:1]\notin X \]to the hyperplane \(H=V(x_n)\cong\PP^{n-1}\) restricts to a morphism\[ \pi:X\longrightarrow X'\coloneqq\overline{\pi(X)}\subseteq H. \]
Projection from \(P\) is \[ [x_0:\cdots:x_n]\longmapsto[x_0:\cdots:x_{n-1}]. \] Its only indeterminacy point is \(P\). Since \(P\notin X\), its restriction to \(X\) is everywhere defined and is a morphism, by the coordinate description of projection in Exercise I.3.14. The image of the irreducible space \(X\) is irreducible, so its closure \(X'\) is a projective variety.
3. The induced map of function fields \(k(X')\hookrightarrow K\) is an isomorphism.
On the dense open set \(x_0\ne0\), the projection retains the affine coordinate functions \[ u_i=x_i/x_0\qquad(1\le i\le n-1). \] Therefore its pullback identifies \(k(X')\) with the subfield \[ k(u_1,\ldots,u_{n-1})\subseteq K. \] Because \(n\ge r+2\), one has \(r+1\le n-1\). Step 1 shows that the first \(r+1\) of these retained functions already generate \(K\). Hence \[ k(X')=k(u_1,\ldots,u_{n-1})=K. \]
4. The morphism \(\pi:X\to X'\) is birational.
A dominant morphism of varieties is birational exactly when it induces an isomorphism of function fields [@Har10a, Proposition I.4.4]. Dominance holds by the definition of \(X'\) as the closure of the image, and step 3 gives the required function-field isomorphism. Thus the chosen projection is birational onto its image.
5. Repeating the construction reduces the ambient projective dimension to \(r+1\).
If the current ambient dimension is greater than \(r+1\), step 4 replaces the variety birationally by one in a projective space of dimension one less. Its dimension remains \(r\), because birational varieties have the same function field and therefore the same transcendence degree over \(k\). After finitely many repetitions the image lies in \(\PP^{r+1}\), where an \(r\)-dimensional projective variety has codimension one and is therefore a hypersurface by P-AGH28HYPERSURFACE. This proves the final assertion in the statement.
6. Q.E.D.
Steps <1>1–<1>4 construct the required point, projection, and birational morphism; step 5 gives the stated finite iteration to a hypersurface.