If \(X\) is a nonsingular variety of dimension \(r\) in \(\PP^n\), and if \(n>2r+1\), show that there is a point \(O \notin X\), such that the projection from \(O\) induces a closed immersion of \(X\) into \(\PP^{n-1}\).
If \(X\) is the Veronese surface in \(\PP^5\), which is the 2-uple embedding of \(\PP^2\) (I, Ex. 2.13), show that each point of every secant line of \(X\) lies on infinitely many secant lines. Therefore, the secant variety of \(X\) has dimension 4, and so in this case there is a projection which gives a closed immersion of \(X\) into \(\PP^4\) (II, Ex. 7.7).
A theorem of Severi states that the Veronese surface is the only surface in \(\PP^5\) for which there is a projection giving a closed immersion into \(\PP^4\). Usually one obtains a finite number of double points with transversal tangent planes.
Solution
1. Let\[ \Sigma_{\mathrm{sec}} \]be the union of secant lines\[ \overline{PQ}, \qquad P,Q\in X,\quad P\ne Q. \]Its closure in \(\PP^n\) has dimension at most\[ 2r+1. \]
Over \[ (X\times X)\setminus\Delta \] consider the incidence variety \[ \mathcal S = \{(P,Q,O):O\in\overline{PQ}\}. \] For every ordered pair \((P,Q)\), the fibre is the projective line \(\overline{PQ}\). Hence \[ \dim\mathcal S = 2r+1. \] The image of \(\mathcal S\) is the union of secant lines, so the dimension of its closure satisfies \[ \dim\overline{\Sigma_{\mathrm{sec}}} \le 2r+1. \]
2. Let\[ \Sigma_{\mathrm{tan}} = \bigcup_{P\in X}T_PX \]be the union of the embedded projective tangent spaces. Then\[ \dim\overline{\Sigma_{\mathrm{tan}}}\le2r. \]
Since \(X\) is nonsingular of dimension \(r\), each embedded tangent space \[ T_PX \subseteq \PP^n \] is a projective \(r\)-plane. The tangent-space incidence \[ \mathcal T = \{(P,O)\in X\times\PP^n:O\in T_PX\} \] is the projectivization of a rank-\((r+1)\) vector bundle over \(X\) and hence has dimension \[ r+r=2r. \] Its image is \(\Sigma_{\mathrm{tan}}\), giving the stated bound.
3. There is a point\[ O\in\PP^n \setminus \left( X \cup \overline{\Sigma_{\mathrm{sec}}} \cup \overline{\Sigma_{\mathrm{tan}}} \right). \]
By hypothesis \[ n>2r+1. \] Steps <1>1–<1>2 show that the two closed secant and tangent loci have dimension strictly smaller than \(n\), and \[ \dim X=r<n. \] Their finite union is therefore a proper closed subset of the irreducible space \(\PP^n\). Choose \(O\) in its complement.
4. Projection from the point \(O\) in step 3 separates points of \(X\).
Let \[ \pi_O:X\longrightarrow\PP^{n-1} \] be the projection from \(O\). It is defined everywhere because \(O\notin X\).
3. Hence \(\pi_O\) is injective on closed points, which isthe point-separation condition for the corresponding subsystem of\(H^0(X,\mco_X(1))\).
5. Projection from \(O\) separates tangent vectors of \(X\).
Fix \[ P\in X. \] The differential of linear projection \[ d\pi_O:T_PX_{\mathrm{aff}}\longrightarrow T_{\pi_O(P)}\PP^{n-1} \] has a nonzero kernel exactly when the projective center \(O\) lies in the embedded projective tangent space \[ T_PX. \] Indeed, the kernel direction is the infinitesimal direction of the line from \(P\) toward the center of projection.
Step 3 gives \[ O\notin T_PX \] for every \(P\). Thus every tangent map is injective, so the projected linear system separates tangent vectors.
6. The projection\[ \pi_O:X\longrightarrow\PP^{n-1} \]is a closed immersion.
Projection from \(O\) is the morphism defined by the base-point-free codimension-one subsystem of hyperplane sections consisting of hyperplanes through \(O\). Steps <1>4–<1>5 show that this system separates points and tangent vectors. The closed-immersion criterion for a linear system therefore gives \[ \boxed{\pi_O\text{ is a closed immersion}.} \] This proves (a).
7. Now let\[ \nu_2:\PP^2\longrightarrow X\subseteq\PP^5 \]be the quadratic Veronese embedding. Every secant line to \(X\) is containedin the plane of a Veronese conic.
Let a secant line join the two distinct points \[ \nu_2(P),\nu_2(Q), \qquad P,Q\in\PP^2. \] Let \[ \ell=\overline{PQ}\subseteq\PP^2. \] The restriction \[ \nu_2|_\ell:\ell\cong\PP^1\longrightarrow\PP^5 \] is the complete quadratic Veronese embedding of \(\PP^1\). Its image \[ C=\nu_2(\ell) \] is a nonsingular plane conic spanning a plane \[ \Pi\cong\PP^2. \] Both endpoints of the secant lie on \(C\), so their joining line lies in \(\Pi\).
8. Every point of every secant line of \(X\) lies on infinitely manysecant lines of \(X\).
7. Then\[ O\in\Pi. \]Consider the pencil of all lines in \(\Pi\) through \(O\).
If \(O\in C\), every member except the unique tangent line at the smooth point \(O\) meets \(C\) in \(O\) and one further distinct point.
Suppose \(O\notin C\). Projection from \(O\) gives a finite morphism \[ C\longrightarrow\PP^1 \] of degree \(2\), where the target is the pencil of lines through \(O\). The original secant line from step 7 is one fibre containing two distinct points. Hence this degree-two morphism is generically separable: a purely inseparable degree-two map would be radicial and could not have such a fibre. Therefore the geometric generic fibre consists of two distinct points, and the same holds over a nonempty open subset of the pencil.
In either case infinitely many lines through \(O\) are secants of the conic \(C\). Since \[ C\subseteq X, \] they are also secant lines of the Veronese surface \(X\).
9. The secant variety of the Veronese surface has dimension at most \(4\).
Let \[ \mathcal J = \{(P,Q,O): P,Q\in X,\ P\ne Q,\ O\in\overline{PQ}\}. \] Since \[ \dim X=2, \] one has \[ \dim\mathcal J=2+2+1=5. \] Its image in \(\PP^5\) is dense in the secant variety \[ \operatorname{Sec}(X). \]
By step 8, every point of this image lies on infinitely many secant lines. Thus every fibre of \[ \mathcal J\longrightarrow\operatorname{Sec}(X) \] over a point of the dense image has infinitely many closed points. A zero-dimensional finite-type scheme over an algebraically closed field has only finitely many closed points, so these fibres have dimension at least \(1\). The fibre-dimension inequality therefore gives \[ \dim\operatorname{Sec}(X) \le 5-1 =4. \]
10. The secant variety has dimension at least \(4\).
Use the standard coordinates \[ \nu_2([x:y:z]) = [x^2:y^2:z^2:xy:xz:yz]. \] Near \[ P_0=[1:0:0], \qquad Q_0=[0:1:0], \] write \[ P(a,b)=[1:a:b], \qquad Q(c,d)=[c:1:d]. \] Restrict to the four-parameter family with \(c=0\) and consider \[ \Psi(a,b,d,t) = [\nu_2(P(a,b))+t\,\nu_2(Q(0,d))]. \] On the target chart where the \(x^2\)-coordinate is nonzero, normalize that coordinate to \(1\). The image coordinates are then \[ \begin{aligned} A&=a^2+t,\\ B&=b^2+td^2,\\ C&=a,\\ D&=b,\\ E&=ab+td. \end{aligned} \] On the open subset \[ t\ne0, \] these equations recover the four parameters rationally: \[ a=C, \qquad b=D, \qquad t=A-C^2, \qquad d=\frac{E-CD}{A-C^2}. \] Thus \(\Psi\) is generically one-to-one onto its image on this four-dimensional open set. Its image therefore has dimension \(4\). Since that image is contained in the secant variety, \[ \dim\operatorname{Sec}(X)\ge4. \]
11. The secant variety of the Veronese surface has dimension exactly\[ \boxed{4}. \]
Steps <1>9–<1>10 give the opposite inequalities \[ \dim\operatorname{Sec}(X)\le4 \qquad\text{and}\qquad \dim\operatorname{Sec}(X)\ge4. \] Therefore equality holds.
12. There is a projection\[ X\hookrightarrow\PP^4 \]which is a closed immersion.
By step 11, \[ \operatorname{Sec}(X) \subsetneq \PP^5. \] Choose \[ O\in\PP^5\setminus\operatorname{Sec}(X). \] The closure of the secant variety contains the embedded tangent spaces of the smooth surface \(X\): every tangent direction is a limit of secant directions, and the projective tangent plane is the union of its tangent lines through the point. Thus \(O\) lies on neither a secant line nor an embedded tangent plane of \(X\).
The point- and tangent-separation argument of steps <1>4–<1>6 therefore applies to projection from \(O\) and gives a closed immersion \[ \boxed{X\hookrightarrow\PP^4}. \] This is the projection asserted in (b).
13. Q.E.D.
Steps <1>1–<1>6 prove (a). Steps <1>7–<1>8 prove that each point on every secant line lies on infinitely many secants; steps <1>9–<1>11 compute the secant-variety dimension; and step 12 gives the closed immersion into \(\PP^4\) required in (b).