## I.7: Intersections in Projective Space ### 7.1. #to-work 1. Find the degree of the $d$-uple embedding of $\mathbf{P}^n$ in $\mathbf{P}^N\left(\right.$ Ex. 2.12).[^answer_one] 2. Find the degree of the Segre embedding of $\mathbf{P}^r \times \mathbf{P}^s$ in $\mathbf{P}^{\prime}($ Ex. 2.14).[^answer_two] [^answer_one]: Answer: $d^n$. [^answer_two]: Answer: ${r+s\choose s}$. ### 7.2. #to-work Let $Y$ be a variety of dimension $r$ in $\mathbf{P}^n$, with Hilbert polynomial $P_Y$. We define the arithmetic genus of $Y$ to be $p_a(Y) = (-1)^r\left(P_Y(0)-1\right)$. This is an important invariant which (as we will see later in (III, Ex. 5.3)) is independent of the projective embedding of $Y$. 1. Show that $p_a\left(\mathbf{P}^n\right)=0$. 2. If $Y$ is a plane curve of degree $d$, show that $p_a(Y)=\frac{1}{2}(d-1)(d-2)$. 3. More generally, if $H$ is a hypersurface of degree $d$ in $\mathbf{P}^n$, then $p_a(H)={d-1\choose n}$. 4. If $Y$ is a complete intersection (Ex. 2.17) of surfaces of degrees $a, h$ in $\mathbf{P}^3$, then $p_a(Y)=\frac{1}{2} a b(a+b-4)+1$. (e) Let $Y^r \subseteq \mathbf{P}^n, Z^{s} \subseteq \mathbf{P}^m$ be projective varieties, and embed $Y \times Z \subseteq \mathbf{P}^n \times$ $\mathbf{P}^m \rightarrow \mathbf{P}^N$ by the Segre embedding. Show that $$ p_a(Y \times Z)=p_a(Y) p_a(Z)+(-1)^{s} p_a(Y)+(-1)^r p_a(Z) . $$ ### 7.3. The Dual Curve. #to-work Let $Y \subseteq \mathbf{P}^2$ be a curve. We regard the set of lines in $\mathbf{P}^2$ as another projective space, $\left(\mathbf{P}^2\right)^*$. by taking $(a_0, a_1, a_2)$ as homogeneous coordinates of the line $L: a_0 x_0+a_1 x_1+a_2 x_2=0$. For each nonsingular point $P \in Y$, show that there is a unique line $T_P(Y)$ whose intersection multiplicity with $Y$ at $P$ is $>1$. This is the tangent line to $Y$ at $P$. Show that the mapping $P \mapsto T_P(Y)$ defines a morphism of Reg $Y$ (the set of nonsingular points of $Y)$ into $\left(\mathbf{P}^2\right)^*$. The closure of the image of this morphism is called the dual curve $Y^* \subseteq\left(\mathbf{P}^2\right)^*$ of $Y$. ### 7.4. #to-work Given a curve $Y$ of degree $d$ in $\mathbf{P}^2$, show that there is a nonempty open subset $U$ of $\left(\mathbf{P}^2\right)^*$ in its Zariski topology such that for each $L \in U, L$ meets $Y$ in exactly $d$ points. [^hint_7.4] This result shows that we could have defined the degree of $Y$ to be the number $d$ such that almost all lines in $\mathbf{P}^2$ meet $Y$ in $d$ points, where "almost all" refers to a nonempty open set of the set of lines, when this set is identified with the dual projective space $\left(\mathbf{P}^2\right)^*$ [^hint_7.4]: Hint: Show that the set of lines in $\left(\mathbf{P}^2\right)^*$ which are either tangent to $Y$ or pass through a singular point of $Y$ is contained in a proper closed subset. ### 7.5. #to-work 1. Show that an irreducible curve $Y$ of degree $d>1$ in $\mathbf{P}^2$ cannot have a point of multiplicity $\geqslant d($ Ex. 5.3). 2. If $Y$ is an irreducible curve of degree $d>1$ having a point of multiplicity $d-1$. then $Y$ is a rational curve (Ex. 6.1). ### 7.6. Linear Varieties. #to-work Show that an algebraic set $Y$ of pure dimension $r$ (i.e., every irreducible component of $Y$ has dimension $r$ ) has degree 1 if and only if $Y$ is a linear variety (Ex. 2.11).[^hint_7.6] [^hint_7.6]: Hint: First, use (7.7) and treat the case $\operatorname{dim} Y=1$. Then do the general case by cutting with a hyperplane and using induction. ### 7.7. #to-work Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $\mathbf{P}^n$. Let $P \in Y$ be a nonsingular point. Define $X$ to be the closure of the union of all lines $P Q$, where $Q \in Y, Q \neq P$. 1. Show that $X$ is a variety of dimension $r+1$. 2. Show that $\operatorname{deg} X