Let \(\PP^n\) be a hyperplane in \(\PP^{n+1}\) and let \(P \in \PP^{n+1} \sm \PP^n\). Define \(\phi: \PP^{n+1} \sm \ts{P} \to \PP^n\) by letting \(\phi(Q)\) be the intersection with \(\PP^n\) of the unique line through \(P\) and \(Q\).
Show that \(\phi\) is a morphism.
Let \(Y \subseteq \PP^3\) be the twisted cubic curve, the image of the \(3\)-uple embedding of \(\PP^1\). If \(t, u\) are the homogeneous coordinates on \(\PP^1\), then \(Y\) is given parametrically by \((x,y,z,w) = (t^3, t^2 u, t u^2, u^3)\). Let \(P = \tv{0 : 0 : 1 : 0}\) and let \(\PP^2\) be the hyperplane \(z = 0\). Show that the projection of \(Y\) from \(P\) is a cuspidal cubic curve in the plane, and find its equation.
Solution
1. After a projective linear change of coordinates, part (a) reduces to\[ P=[0:\cdots:0:1], \qquad \PP^n=Z(x_{n+1})\subseteq\PP^{n+1}. \]
Choose a nonzero vector representing \(P\) and a basis of the vector hyperplane whose projectivization is the given \(\PP^n\). Since \(P\) does not lie in that hyperplane, adjoining the representative of \(P\) gives a basis of the ambient vector space. The resulting linear automorphism of the vector space induces a projective automorphism carrying the pair \((P,\PP^n)\) to the displayed coordinate pair. Conjugating the projection by projective automorphisms preserves the property of being a morphism.
2. In these coordinates,\[ \phi([a_0:\cdots:a_n:a_{n+1}])=[a_0:\cdots:a_n]. \]This is a morphism on \(\PP^{n+1}\setminus\{P\}\), proving (a).
For \(Q=[a_0:\cdots:a_{n+1}]\ne P\), not all of \(a_0,\ldots,a_n\) vanish. The line through \(P\) and \(Q\) consists of the projective points represented by \[ \lambda(a_0,\ldots,a_n,a_{n+1})+\mu(0,\ldots,0,1). \] Choosing \(\mu=-\lambda a_{n+1}\) gives its unique intersection with \(x_{n+1}=0\), namely \[ [a_0:\cdots:a_n:0]. \] After identifying that hyperplane with \(\PP^n\), this is the displayed formula.
The coordinate functions \(x_0,\ldots,x_n\) are homogeneous of the same degree and have no common zero on the domain \(\PP^{n+1}\setminus\{P\}\). Hence they define a morphism there [@Har10a, Chapter I, §3].
3. For the data in part (b), projection from \(P=[0:0:1:0]\) onto \(z=0\) restricts to\[ [t:u]\longmapsto[t^3:t^2u:u^3] \]in the plane coordinates \([x:y:w]\).
Here the coordinate of the center is \(z\). The same calculation as in step 2 therefore deletes the \(z\)-coordinate: \[ [x:y:z:w]\longmapsto[x:y:w]. \] Substituting the twisted-cubic parametrization \[ [x:y:z:w]=[t^3:t^2u:tu^2:u^3] \] gives the displayed parametrization. The center \(P\) is not on \(Y\), so this restriction is defined everywhere on \(Y\).
4. The image in \(\PP^2\) is exactly the cubic\[ C=Z(y^3-x^2w). \]
Every parameterized point satisfies \[ (t^2u)^3=(t^3)^2u^3, \] so the image is contained in \(C\).
Conversely, let \([x:y:w]\in C\). If \(x\ne0\), normalize to \(x=1\). The equation becomes \(w=y^3\), and the point is \[ [1:y:y^3], \] which is the image of \([1:y]\). If \(x=0\), then \(y^3=0\), so \(y=0\) and the point is \([0:0:1]\), the image of \([0:1]\). Thus every point of \(C\) occurs.
5. The cubic \(C\) is cuspidal at \([0:0:1]\).
On the affine chart \(w=1\) around \([0:0:1]\), its equation is \[ y^3=x^2, \] with parametrization \[ x=t^3, \qquad y=t^2. \] The origin is singular: for \(F=y^3-x^2\), both partial derivatives vanish there in every characteristic. It is the only singular point of the projective cubic \(y^3-x^2w=0\): the equations \[ F=0, \qquad F_x=-2xw=0, \qquad F_y=3y^2=0, \qquad F_w=-x^2=0 \] force \(x=y=0\), also in characteristics \(2\) and \(3\). The lowest-degree term of the affine equation at the origin is \(-x^2\), so the tangent cone is the doubled line \(x=0\); together with the one-parameter normalization \(t\mapsto(t^3,t^2)\), this is the standard cuspidal cubic.
6. Q.E.D.
Steps <1>1–<1>2 prove (a), and steps <1>3–<1>5 prove (b), with plane equation \[ \boxed{y^3=x^2w}. \]
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P-AGH212DUPLEThe $d$-uple embedding of $\PP^n$ in $\PP^N$uses
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P-AGH44RATCURVESConics, the cuspidal cubic, and the nodal cubic are rational curvesuses