1_2x

I.2: Projective Varieties

2.1 #to_work

Prove the “homogeneous Nullstellensatz,” which says if \(a \subseteq S\) is a homogeneous ideal, and if \(f \in S\) is a homogeneous polynomial with deg \(f>0\), such that \(f(P)=0\) for all \(P \in Z(a)\) in \(\mathbf{P}^n\), then \(f^u \in a\) for some \(q>0\). ¹

2.2 #to_work

For a homogeneous ideal \(a \subseteq S\), show that the following conditions are equivalent:

• \(Z(a) = \varnothing\) (the empty set);
• \(\sqrt{a}=\) either \(S\) or the ideal \(S_{+}=\bigoplus_{d>0} S_d\);
• \(a \supseteq S_d\) for some \(d>0\).

2.3 #to_work

• If \(T_1 \subseteq T_2\) are subsets of \(S^h\). then \(Z\left(T_1\right) \supseteq Z\left(T_2\right)\).
• If \(Y_1 \subseteq Y_2\) are subsets of \(\mathbf{P}^n\), then \(I\left(Y_1\right) \supseteq I\left(Y_2\right)\).

• For any two subsets \(Y_1, Y_2\) of \(\mathbf{P}^n, I\left(Y_1 \cup Y_2\right)=I\left(Y_1\right) \cap I\left(Y_2\right)\).
• If \(a \subseteq S\) is a homogeneous ideal with \(Z(a) \neq \varnothing\). then \(I(Z(a))=\sqrt a\).

• For any subset \(Y \subseteq \mathbf{P}^n, Z(I(Y))=\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu\).

2.4 #to_work

• There is a 1-1 inclusion-reversing correspondence between algebraic sets in \(\mathbf{P}^n\). and homogeneous radical ideals of \(S\) not equal to \(S_{+}\) given by \(Y \mapsto I(Y)\) and \(a \mapsto Z(a)\). ²
• An algebraic set \(Y \subseteq \mathbf{P}^n\) is irreducible if and only if \(I\left(Y^{\prime}\right)\) is a prime ideal.

• Show that \(\mathbf{P}^n\) itself is irreducible.

2.5 #to_work

• \(\mathbf{P}^n\) is a noetherian topological space.
• Every algebraic set in \(\mathrm{P}^n\) can be written uniquely as a finite union of irreducible algebraic sets. no one containing another. These are called its irreducible components.

2.6 #to_work

If \(Y\) is a projective variety with homogeneous coordinate ring \(S(Y)\), show that \(\operatorname{dim} S(Y)=\operatorname{dim} Y+1\). ³

2.7 #to_work

• \(\operatorname{dim} \mathbf{P}^n=n\).
• If \(Y \subseteq \mathbf{P}^n\) is a quasi-projective variety, then \(\operatorname{dim} Y=\operatorname{dim} \mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu\). ⁴

2.8 #to_work

A projective variety \(Y \subseteq \mathbf{P}^n\) has dimension \(n-1\) if and only if it is the zero set of a single irreducible homogeneous polynomial \(f\) of positive degree. \(Y\) is called a hypersurface in \(\mathbf{P}^n\).

2.9 Projective Closure of an Affine Variety #to_work

If \(Y \subseteq \mathbf{A}^n\) is an affine variety, we identify \(\mathbf{A}^n\) with an open set \(U_0 \subseteq \mathbf{P}^n\) by the homeomorphism \(\varphi_0\). Then we can speak of \(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu\), the closure of \(Y\) in \(\mathbf{P}^n\), which is called the projective closure of \(Y\).

• Show that \(I(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu)\) is the ideal generated by \(\beta(I(Y))\), using the notation of the proof of \((2.2)\).
• Let \(Y \subseteq \mathbf{A}^3\) be the twisted cubic of (Ex. 1.2). Its projective closure \(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu \subseteq \mathbf{P}^3\) is called the twisted cubic curve in \(\mathbf{P}^3\). Find generators for \(I(Y)\) and \(I(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu)\), and use this example to show that if \(f_1, \ldots, f_r\) generate \(I(Y)\), then \(\beta\left(f_1\right), \ldots, \beta\left(f_r\right)\) do not necessarily generate \(I(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu)\).

2.10 The Cone Over a Projective Variety (Fig. 1) #to_work

Let \(Y \subseteq \mathbf{P}^n\) be a nonempty algebraic set, and let \(\theta: \mathbf{A}^{n+1}-\{(0, \ldots, 0)\} \rightarrow \mathbf{P}^n\) be the map which sends the point with affine coordinates \(\left(a_0, \ldots, a_n\right)\) to the point with homogeneous coordinates \(\left(a_0, \ldots, a_n\right)\). We define the affine cone over \(Y\) to be \begin{align*} C(Y)=\theta^{-1}(Y) \cup\{(0, \ldots, 0)\} . \end{align*}

• Show that \(C(Y)\) is an algebraic set in \(\mathbf{A}^{n+1}\), whose ideal is equal to \(I(Y)\), considered as an ordinary ideal in \(k\left[x_0, \ldots, x_n\right]\).
• \(C(Y)\) is irreducible if and only if \(Y\) is.

• \(\operatorname{dim} C(Y)=\operatorname{dim} Y+1\).

Sometimes we consider the projective closure \(\overline{C(Y)}\) of \(C(Y)\) in \(\mathbf{P}^{n+1}\). This is called the projective cone over \(Y\).

2.11 Linear Varieties in \(\mathbf{P}^n\) #to_work

A hypersurface defined by a linear polynomial is called a hyperplane.

• Show that the following two conditions are equivalent for a variety \(Y\) in \(\mathbf{P}^n\) :

  • \(I(Y)\) can be generated by linear polynomials.
  • \(Y\) can be written as an intersection of hyperplanes.

  In this case we say that \(Y\) is a linear variety in \(\mathbf{P}^n\).

• If \(Y\) is a linear variety of dimension \(r\) in \(\mathbf{P}^n\), show that \(I(Y)\) is minimally generated by \(n-r\) linear polynomials.

• Let \(Y, Z\) be linear varieties in \(\mathbf{P}^n\), with \(\operatorname{dim} Y=i, \operatorname{dim} Z=\) s. If \(r+s-n \geqslant 0\), then \(Y \cap Z \neq \varnothing\). Furthermore, if \(Y \cap Z \neq \varnothing\), then \(Y \cap Z\) is a linear variety of dimension \(\geqslant r+s-n\). ⁵

2.12 The \(d\)-uple Embedding #to_work

For given \(n, d>0\), let \(M_0, M_1, \ldots, M_N\) be all the monomials of degree \(d\) in the \(n+1\) variables \(x_0, \ldots, x_n\), where \(N={n+d\choose n}-1\). We define a mapping \(\rho_d: \mathbf{P}^n \rightarrow \mathbf{P}^{N}\) by sending the point \(P=\left(a_0, \ldots, a_n\right)\) to the point \(\rho_d(P)=\left(M_0(a), \ldots, M_N(a)\right)\) obtained by substituting the \(a_t\) in the monomials \(M_J\). This is called the \(d\)-uple embedding of \(\mathbf{P}^n\) in \(\mathbf{P}^N\). For example, if \(n=1, d=2\), then \(N=2\), and the image \(Y\) of the 2-uple embedding of \(\mathbf{P}^1\) in \(\mathbf{P}^2\) is a conic.

• Let \(\theta: k\left[y_0, \ldots, y_v\right] \rightarrow k\left[x_0, \ldots, x_n\right]\) be the homomorphism defined by sending \(y_i\) to \(M_i\), and let a be the kernel of \(\theta\). Then \(a\) is a homogeneous prime ideal, and so \(Z\) (a) is a projective variety in \(\mathbf{P}^{N}\).
• Show that the image of \(\rho_d\) is exactly \(Z(a)\). ⁶

• Now show that \(\rho_d\) is a homeomorphism of \(\mathbf{P}^n\) onto the projective variety \(Z\) (a).
• Show that the twisted cubic curve in \(\mathbf{P}^3\) (Ex. 2.9) is equal to the 3-uple embedding of \(\mathbf{P}^1\) in \(\mathbf{P}^3\), for suitable choice of coordinates.

2.13 #to_work

Let \(Y\) be the image of the 2-uple embedding of \(\mathbf{P}^2\) in \(\mathbf{P}^5\). This is the Veronese surface. If \(Z \subseteq Y\) is a closed curve (a curve is a variety of dimension 1), show that there exists a hypersurface \(V \subseteq \mathbf{P}^5\) such that \(V \cap Y = Z\).

2.14 The Segre Embedding #to_work

Let \(\psi: \mathbf{P}^r \times \mathbf{P}^{s} \rightarrow \mathbf{P}^{N}\) be the map defined by sending the ordered pair \(\left(a_0, \ldots, a_r\right) \times\left(b_0, \ldots, b_s\right)\) to \(\left(\ldots, a_i b_j, \ldots\right)\) in lexicographic order. where \(N=r s+r+s\). Note that \(\psi\) is well-defined and injective. It is called the Segre embedding. Show that the image of \(\psi\) is a subvariety of \(\mathbf{P}^N\). ⁷

2.15 The Quadric Surface in \(\mathbf{P}^3\) (Fig. 2) #to_work

Consider the surface \(Q\) (a surface is a variety of dimension 2) in \(\mathbf{P}^3\) defined by the equation \(x y-zw =0\).

• Show that \(Q\) is equal to the Segre embedding of \(\mathbf{P}^1 \times \mathbf{P}^1\) in \(\mathbf{P}^3\). for suitable choice of coordinates.
• Show that \(Q\) contains two families of lines (a line is a linear variety of dimension 1) \(\left\{L_t\right\},\left\{{M_t}\right\}\), each parametrized by \(t \in \mathbf{P}^1\). with the properties that if \(L_t \neq L_u\). then \(L_t \cap L_u=\varnothing\) : if \(M_t \neq M_u, M_t \cap M_u=\varnothing\), and for all \(t,u\), \(L_t \cap M_u=\) one point.

• Show that \(Q\) contains other curves besides these lines, and deduce that the Zariski topology on \(Q\) is not homeomorphic via \(\psi\) to the product topology on \(\mathbf{P}^1 \times \mathbf{P}^1\) (where each \(\mathbf{P}^1\) has its Zariski topology).

2.16 #to_work

• The intersection of two varieties need not be a variety. For example, let \(Q_1\) and \(Q_2\) be the quadric surfaces in \(\mathbf{P}^3\) given by the equations \(x^2-y w=0\) and \(x y-z w=0\), respectively. Show that \(Q_1 \cap Q_2\) is the union of a twisted cubic curve and a line.
• Even if the intersection of two varieties is a variety, the ideal of the intersection may not be the sum of the ideals. For example, let \(C\) be the conic in \(\mathbf{P}^2\) given by the equation \(xy-zw=0\). Let \(L\) be the line given by \(y=0\). Show that \(C \cap L\) consists of one point \(P\), but that \(I(C)+I(L) \neq I(P)\).

2.17 Complete intersections #to_work

A variety \(Y\) of dimension \(r\) in \(\mathbf{P}^n\) is a (strict) complete intersection if \(I(Y)\) can be generated by \(n-r\) elements. \(Y\) is a set-theoretic complete intersection if \(Y\) can be written as the intersection of \(n-r\) hypersurfaces.

• Let \(Y\) be a variety in \(\mathbf{P}^n\), let \(Y=Z(a)\); and suppose that a can be generated by \(q\) elements. Then show that \(\operatorname{dim} Y \geqslant n-q\).
• Show that a strict complete intersection is a set-theoretic complete intersection.

• * The converse of (b) is false. For example let \(Y\) be the twisted cubic curve in \(\mathbf{P}^3\) (Ex. 2.9). Show that \(I(Y)\) cannot be generated by two elements. On the other hand, find hypersurfaces \(\mathrm{H}_1, \mathrm{H}_2\) of degrees 2,3 respectively, such that \(Y=H_1 \cap H_2\).
• ** It is an unsolved problem whether every closed irreducible curve in \(\mathbf{P}^3\) is a set-theoretic intersection of two surfaces. See Hartshorne \([1]\) and Hartshorne \([5.III, \text{section } 5]\) for commentary.