4_3x

IV.3: Embeddings in Projective Space

II.3.1. #to_work

If \(X\) is a curve of genus 2 , show that a divisor \(D\) is very ample \(\Leftrightarrow \operatorname{deg} D \geqslant 5\). This strengthens (3.3.4).

II.3.2. #to_work

Let \(X\) be a plane curve of degree 4 .

Show that the effective canonical divisors on \(X\) are exactly the divisors \(X.L\), where \(L\) is a line in \(\mathbf{P}^2\).
If \(D\) is any effective divisor of degree 2 on \(X\), show that \(\operatorname{dim}|D|=0\).

Conclude that \(X\) is not hyperelliptic (Ex. 1.7).

If \(X\) is a curve of genus \(\geqslant 2\) which is a complete intersection (II, Ex. 8.4) in some \(\mathbf{P}^n\), show that the canonical divisor \(K\) is very ample. Conclude that a curve of genus 2 can never be a complete intersection in any \(\mathbf{P}^n\). Cf. (Ex. 5.1).

II.3.4. The Rational Normal Curve. #to_work

Let \(X\) be the \(d\)-uple embedding (I, Ex. 2.12) of \(\mathbf{P}^1\) in \(\mathbf{P}^d\), for any \(d \geqslant 1\). We call \(X\) the rational normal curve of degree \(d\) in \(\mathbf{P}^d\).

Show that \(X\) is projectively normal, and that its homogeneous ideal can be generated by forms of degree 2 .
If \(X\) is any curve of degree \(d\) in \(\mathbf{P}^n\), with \(d \leqslant n\), which is not contained in any \(\mathbf{P}^{n-1}\), show that in fact \(d=n, g(X)=0\), and \(X\) differs from the rational normal curve of degree \(d\) only by an automorphism of \(\mathbf{P}^d\). Cf. (II. 7.8.5).

In particular, any curve of degree 2 in any \(\mathbf{P}^n\) is a conic in some \(\mathbf{P}^2\).
A curve of degree 3 in any \(\mathbf{P}^n\) must be either a plane cubic curve, or the twisted cubic curve in \(\mathbf{P}^3\).

II.3.5. #to_work

Let \(X\) be a curve in \(\mathbf{P}^3\), which is not contained in any plane.

If \(O \notin X\) is a point, such that the projection from \(O\) induces a birational morphism \(\varphi\) from \(X\) to its image in \(\mathbf{P}^2\), show that \(\varphi(X)\) must be singular. ¹
If \(X\) has degree \(d\) and genus \(g\), conclude that \(g<\frac{1}{2}(d-1)(d-2)\). (Use (Ex. 1.8).)

Now let \(\left\{X_t\right\}\) be the flat family of curves induced by the projection (III, 9.8.3) whose fibre over \(t=1\) is \(X\), and whose fibre \(X_0\) over \(t=0\) is a scheme with support \(\varphi(X)\). Show that \(X_0\) always has nilpotent elements. Thus the example (III, 9.8.4) is typical.

II.3.6. #to_work

Curves of Degree 4.

If \(X\) is a curve of degree 4 in some \(\mathbf{P}^n\), show that either
- \(g=0\), in which case \(X\) is either the rational normal quartic in \(\mathbf{P}^4\) (Ex. 3.4) or the rational quartic curve in \(\mathbf{P}^3\) (II, 7.8.6), or
- \(X \subseteq \mathbf{P}^2\), in which case \(g=3\), or
- \(X \subseteq \mathbf{P}^3\) and \(g=1\).
In the case \(g=1\), show that \(X\) is a complete intersection of two irreducible quadric surfaces in \(\mathbf{P}^3\) (I, Ex. 5.11). ²

II.3.7. #to_work

In view of (3.10), one might ask conversely, is every plane curve with nodes a projection of a nonsingular curve in \(\mathbf{P}^3\) ? Show that the curve \(x y+x^4+y^4=0\) (assume char \(k \neq 2\) ) gives a counterexample.

II.3.8. #to_work

We say a (singular) integral curve in \(\mathbf{P}^n\) is strange if there is a point which lies on all the tangent lines at nonsingular points of the curve.

There are many singular strange curves, e.g., the curve given parametrically by \(x=t, y=t^p, z=t^{2 p}\) over a field of characteristic \(p>0\).
Show, however, that if char \(k=0\), there aren’t even any singular strange curves besides \(\mathbf{P}^1\).

II.3.9. Bertini’s Lemma. #to_work

Prove the following lemma of Bertini: if \(X\) is a curve of degree \(d\) in \(\mathbf{P}^3\), not contained in any plane, then for almost all planes \(H \subseteq \mathbf{P}^3\) (meaning a Zariski open subset of the dual projective space \(\left.\left(\mathbf{P}^3\right)^*\right)\), the intersection \(X \cap H\) consists of exactly \(d\) distinct points, no three of which are collinear.

II.3.10. Not every secant is a multisecant. #to_work

Generalize the statement that “not every secant is a multisecant” as follows. If \(X\) is a curve in \(\mathbf{P}^n\), not contained in any \(\mathbf{P}^{n-1}\), and if char \(k=0\), show that for almost all choices of \(n-1\) points \(P_1, \ldots, P_{n-1}\) on \(X\), the linear space \(L^{n-2}\) spanned by the \(P_i\) does not contain any further points of \(X\).

II.3.11 #to_work

If \(X\) is a nonsingular variety of dimension \(r\) in \(\mathbf{P}^n\), and if \(n>2 r+1\), show that there is a point \(O \notin X\), such that the projection from \(O\) induces a closed immersion of \(X\) into \(\mathbf{P}^{n-1}\).
If \(X\) is the Veronese surface in \(\mathbf{P}^5\), which is the 2-uple embedding of \(\mathbf{P}^2\) (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 \(\mathrm{P}^4\) (II, Ex. 7.7). ³

II.3.12. #to_work

For each value of \(d=2,3,4,5\) and \(r\) satisfying \(0 \leqslant r \leqslant \frac{1}{2}(d-1)(d-2)\), show that there exists an irreducible plane curve of degree \(d\) with \(r\) nodes and no other singularities.

Footnotes

Hint: Calculate \(\operatorname{dim} H^0\left(X, \mathcal{O}_X(1)\right)\) two ways.

Hint: Use the exact sequence \(0 \rightarrow \mathcal{I}_X \rightarrow\) \(\mathcal{O}_{\mathbf{P}^3} \rightarrow \mathcal{O}_X \rightarrow 0\) to compute \(\operatorname{dim} H^0\left(\mathbf{P}^3, \mathcal{I}_X(2)\right)\), and thus conclude that \(X\) is contained in at least two irreducible quadric surfaces.

A theorem of Severi \([1]\) states that the Veronese surface is the only surface in \(\mathbf{P}^5\) for which there is a projection giving a closed immersion into \(\mathbf{P}^4\). Usually one obtains a finite number of double points with transversal tangent planes.