## 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 .

a. Show that the effective canonical divisors on $X$ are exactly the divisors $X.L$, where $L$ is a line in $\mathbf{P}^2$.

b. If $D$ is any effective divisor of degree 2 on $X$, show that $\operatorname{dim}|D|=0$.

c. Conclude that $X$ is not hyperelliptic (Ex. 1.7).

### II.3.3.  #to-work

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 $\mathbf{P}^d$.

a. Show that $X$ is projectively normal, and that its homogeneous ideal can be generated by forms of degree 2 .

b. 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).

c. In particular, any curve of degree 2 in any $\mathbf{P}^n$ is a conic in some $\mathbf{P}^2$.

d. 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.

a. 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.[^hint2.3.5.a]

b. If $X$ has degree $d$ and genus $g$, conclude that $g<\frac{1}{2}(d-1)(d-2)$. (Use (Ex. 1.8).)

c. 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.

[^hint2.3.5.a]: 
Hint: Calculate $\operatorname{dim} H^0\left(X, \mathcal{O}_X(1)\right)$ two ways.

### II.3.6.  #to-work

Curves of Degree 4.

a. If $X$ is a curve of degree 4 in some $\mathbf{P}^n$, show that either
    (1) $g=0$, 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
    (2) $X \subseteq \mathbf{P}^2$, in which case $g=3$, or
    (3) $X \subseteq \mathbf{P}^3$ and $g=1$.

b. 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).[^hint.2.3.6.b]

[^hint.2.3.6.b]: 
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.


### 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.

a. 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 $p>0$.

b. 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


a. 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 $\mathbf{P}^{n-1}$.

b. 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).[^rmk2.3.11.b]

[^rmk2.3.11.b]: 
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.

### 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.