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I.7: Intersections in Projective Space

7.1. #to_work

• Find the degree of the \(d\)-uple embedding of \(\mathbf{P}^n\) in \(\mathbf{P}^N\left(\right.\) Ex. 2.12). ¹

• Find the degree of the Segre embedding of \(\mathbf{P}^r \times \mathbf{P}^s\) in \(\mathbf{P}^{\prime}(\) Ex. 2.14). ²

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 \(Y\).

• Show that \(p_a\left(\mathbf{P}^n\right)=0\).
• If \(Y\) is a plane curve of degree \(d\), show that \(p_a(Y)=\frac{1}{2}(d-1)(d-2)\).
• More generally, if \(H\) is a hypersurface of degree \(d\) in \(\mathbf{P}^n\), then \(p_a(H)={d-1\choose n}\).
• If \(Y\) is a complete intersection (Ex. 2.17) of surfaces of degrees \(a, h\) in \(\mathbf{P}^3\), then \(p_a(Y)=\frac{1}{2} a b(a+b-4)+1\).

• 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 \begin{align*} p_a(Y \times Z)=p_a(Y) p_a(Z)+(-1)^{s} p_a(Y)+(-1)^r p_a(Z) . \end{align*}

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. ³

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 \(\left(\mathbf{P}^2\right)^*\)

7.5. #to_work

• 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).
• 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). ⁴

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 \(Q \in Y, Q \neq P\).

• Show that \(X\) is a variety of dimension \(r+1\).
• Show that \(\operatorname{deg} X<d\). ⁵

7.8. #to_work

Let \(Y^r \subseteq \mathbf{P}^n\) be a variety of degree 2. Show that \(Y\) is contained in a linear subspace \(L\) of dimension \(r+1\) in \(\mathbf{P}^n\). Thus \(Y\) is isomorphic to a quadric hypersurface in \(\mathbf{P}^{r+1}(\mathrm{Ex} .5 .12)\)