Chapter 1

Recall that a curve is rational if it is birationally equivalent to \(\mathbf{P}^{1}\) (Ex. 4.4). Let \(Y\) be a nonsingular rational curve which is not isomorphic to \(\mathbf{P}^{1}\) (a) Show that \(Y\) is isomorphic to an open subset of \(\mathbf{A}^{1}\) (b) Show that \(Y\) is affine. (c) Show that \(A(Y)\) is a unique factorization domain.

(a) Show that any conic in \(\mathbf{A}^{2}\) is isomorphic either to \(\mathbf{A}^{1}\) or \(\mathbf{A}^{1}-; 0\), (cf. Ex. 1.1 ). (b) Show that \(\mathbf{A}^{1}\) is \(n o t\) isomorphic to any proper open subset of itself. (This result is generalized by (Ex. 6.7 ) below.) (c) Any conic in \(\mathbf{P}^{2}\) is isomorphic to \(\mathbf{P}^{1}\) (d) We will see later (Ex. 4.8 ) that any two curves are homeomorphic. But show now that \(\mathbf{A}^{2}\) is not even homeomorphic to \(\mathbf{P}^{2}\) (e) If an affine variety is isomorphic to a projective variety, then it consists of only one point.

(a) Let \(Y\) be the plane curve \(y=x^{2}\) (i.e., \(Y\) is the zero set of the polynomial \(f=\) \(y-x^{2}\) ). Show that \(A(Y)\) is isomorphic to a polynomial ring in one variable over \(k\) (b) Let \(Z\) be the plane curve \(x y=1 .\) Show that \(A(Z)\) is not isomorphic to a polynomial ring in one variable over \(k\) (c) Let \(f\) be any irreducible quadratic polynomial in \(k[x, y],\) and let \(W\) be the conic defined by \(f .\) Show that \(A(W)\) is isomorphic to \(A(Y)\) or \(A(Z) .\) Which one is it when?

If \(f\) and \(g\) are regular functions on open subsets \(U\) and \(V\) of a variety \(X,\) and if \(f=g\) on \(U \cap V\). show that the function which is \(f\) on \(U\) and \(g\) on \(V\) is a regular function on \(U \cup V\). Conclude that if \(f\) is a rational function on \(X\). then there is a largest open subset \(U\) of \(X\) on which \(f\) is represented by a regular function. We say that \(f\) is defined at the points of \(U\).

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 \(Y\). (a) Show that \(p_{a}\left(\mathbf{P}^{n}\right)=0\). (b) If \(Y\) is a plane curve of degree \(d\), show that \(p_{a}(Y)=\frac{1}{2}(d-1)(d-2)\). (c) More generally, if \(H\) is a hypersurface of degree \(d\) in \(\mathbf{P}^{n}\), then \(p_{a}(H)=\left(\begin{array}{c}d-1 \\\ n\end{array}\right)\). (d) If \(Y\) is a complete intersection (Ex. 2.17 ) of surfaces of degrees \(a, b\) in \(\mathbf{P}^{3}\), then \(p_{a}(Y)=\frac{1}{2} a b(a+b-4)+1\). (e) Let \(Y^{\prime} \subseteq \mathbf{P}^{n}, Z^{s} \subseteq \mathbf{P}^{m}\) 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)\\].

For a homogeneous ideal \(a \subseteq S\), show that the following conditions are equivalent: (i) \(Z(a)=\varnothing\) (the empty set): (ii) \(\sqrt{\mathfrak{a}}=\) either \(S\) or the ideal \(S_{+}=\bigoplus_{d>0} S_{d}\) (iii) \(a \geq S_{d}\) for some \(d>0\)

The Twisted Cubic Curve. Let \(Y \subseteq \mathbf{A}^{3}\) be the set \(Y=\left\\{\left(t, t^{2}, t^{3}\right) | t \in k\right\\} .\) Show that \(Y\) is an affine variety of dimension 1. Find generators for the ideal \(I(Y) .\) Show that \(A(Y)\) is isomorphic to a polynomial ring in one variable over \(k .\) We say that \(Y\) is given by the parametric representation \(x=t, y=t^{2}, z=t^{3}\)

The Dual Curve. 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 \(\left(a_{0}, a_{1}, a_{2}\right)\) 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 \(Y\).

(a) Let \(\varphi: X \rightarrow Y\) be a morphism. Then for each \(P \in X . \varphi\) induces a homomorphism of local rings \(\varphi_{p}^{*}: c_{m, p, 1} \rightarrow\left(p_{1}, 1\right.\) (b) Show that a morphism \(\varphi\) is an isomorphism if and only if \(\varphi\) is a homeomorphism, and the induced map \(\varphi_{P}^{*}\) on local rings is an isomorphism, for all \(P \in X\) (c) Show that if \(\varphi(X)\) is dense in \(Y\), then the map \(\varphi_{p}^{*}\) is injective for all \(P \in X\)

Let \(Y\) be the algebraic set in \(\mathbf{A}^{3}\) defined by the two polynomials \(x^{2}-y z\) and \(x z-x .\) Show that \(Y\) is a union of three irreducible components. Describe them and find their prime ideals.

(a) Let \(f\) be the rational function on \(\mathbf{P}^{2}\) given by \(f=x_{1} / x_{0} .\) Find the set of points where \(f\) is defined and describe the corresponding regular function. (b) Now think of this function as a rational map from \(\mathbf{P}^{2}\) to \(\mathbf{A}^{1}\). Embed \(\mathbf{A}^{1}\) in \(\mathbf{P}^{1}\), and let \(\varphi: \mathbf{P}^{2} \rightarrow \mathbf{P}^{1}\) be the resulting rational map. Find the set of points where \(\varphi\) is defined, and describe the corresponding morphism.

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. \(\left[\text { Hint: Show that the set of lines in }\left(\mathbf{P}^{2}\right)^{*} \text { which are either tangent to } Y\) or pass \right. through a singular point of \(Y\) is contained in a proper closed subset.] 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 \(\left(\mathbf{P}^{2}\right)^{*}\).

Let \(Y\) be a nonsingular projective curve. Show that every nonconstant rational function \(f\) on \(Y\) defines a surjective morphism \(\varphi: Y \rightarrow \mathbf{P}^{1},\) and that for every \(P \in \mathbf{P}^{1}\) \(\varphi^{-1}(P)\) is a finite set of points.

Intersection Multiplicity. If \(Y, Z \subseteq \mathbf{A}^{2}\) are two distinct curves, given by equations \(f=0, g=0,\) and if \(P \in Y \cap Z,\) we define the intersection multiplicity \((Y \cdot Z)_{P}\) of \(Y\) and \(\mathcal{L}\) at \(P\) to be the length of the \(\mathscr{O}_{p}\) -module \(\mathscr{C}_{p} f(f, g)\) (a) Show that \((Y \cdot Z)_{P}\) is finite, and \((Y \cdot Z)_{P} \geqslant \mu_{P}(Y) \cdot \mu_{P}(Z)\) (b) If \(P \in Y\), show that for almost all lines \(L\) through \(P\) (i.e., all but a finite number) \((L \cdot Y)_{Y}=\mu_{P}(Y)\) (c) If \(Y\) is a curve of degree \(d\) in \(\mathbf{P}^{2},\) and if \(L\) is a line in \(\mathbf{P}^{2}, L \neq Y,\) show that \((L \cdot Y)=d .\) Here we define \((L \cdot Y)=\sum(L \cdot Y)_{P}\) taken over all points \(P \in\) \(L \cap Y,\) where \((L \cdot Y)_{p}\) is defined using a suitable affine cover of \(\mathbf{P}^{2}\)

(a) 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 \(a \mapsto Z(a) .\) Note: since \(S_{+}\) does not occur in this correspondence, it is sometimes called the irrelercult maximal ideal of \(S\) (b) An algebraic set \(Y \subseteq \mathbf{P}^{n}\) is irreducible if and only if \(I\) ( \(Y\) ') is a prime ideal. (c) Show that \(\mathbf{P}^{n}\) itself is irreducible.

If we identify \(\mathbf{A}^{2}\) with \(\mathbf{A}^{1} \times \mathbf{A}^{1}\) in the natural way, show that the Zariski topology on \(\mathbf{A}^{2}\) is not the product topology of the Zariski topologies on the two copies of \(\mathbf{A}^{1}\)

A variety \(Y\) is rational if it is birationally equivalent to \(\mathbf{P}^{n}\) for some \(n\) (or, equivalently by \((4.5),\) if \(K(Y)\) is a pure transcendental extension of \(k\) ). (a) Any conic in \(\mathbf{P}^{2}\) is a rational curve. (b) The cuspidal cubic \(y^{2}=x^{3}\) is a rational curve. (c) Let \(Y\) be the nodal cubic curve \(y^{2} z=x^{2}(x+z)\) in \(P^{2}\). Show that the projection \(\varphi\) from the point \(P=(0,0,1)\) to the line \(z=0\) (Ex. 3.14 ) induces a birational map from \(Y\) to \(\mathbf{P}^{1}\). Thus \(Y\) is a rational curve.

(a) Show that an irreducible curve \(Y\) of degree \(d>1\) in \(\mathbf{P}^{2}\) cannot have a point of multiplicity \(\geqslant d(\mathrm{Ex} .5 .3)\) (b) 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 ).

Show that a \(k\) -algebra \(B\) is isomorphic to the affine coordinate ring of some algebraic set in \(\mathbf{A}^{n}\), for some \(n\), if and only if \(B\) is a finitely generated \(k\) -algebra with no nilpotent elements.

Linear Varieties. 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: First, use (7.7) and treat the case dim \(Y=1 .\) Then do the general case by cutting with a hyperplane and using induction.]

Automorphisms of \(\mathbf{P}^{1}\). Think of \(\mathbf{P}^{1}\) as \(\mathbf{A}^{1} \cup\\{x\), then we define a fractional linear transformation of \(\mathbf{P}^{1}\) by sending \(x \mapsto(u x+b) /(c x+d),\) for \(u, b, c, d \in k\) \(u d-h c \neq 0\) (a) Show that a fractional linear transformation induces an automorphism of \(\mathbf{P}^{1}\) (i.e., an isomorphism of \(\mathbf{P}^{1}\) with itself). We denote the group of all these fractional linear transformations by PGL(1). (b) Let Aut \(\mathbf{P}^{1}\) denote the group of all automorphisms of \(\mathbf{P}^{1}\). Show that Aut \(\mathbf{P}^{1} \simeq\) Aut \(k(x),\) the group of \(k\) -automorphisms of the field \(k(x)\) (c) Now show that every automorphism of \(k(x)\) is a fractional linear transformation, and deduce that \(P G L(1) \rightarrow\) Aut \(P^{1}\) is an isomorphism.

There are quasi-affine varieties which are not affine. For example, show that \(\mathrm{N}=\mathbf{A}^{2}-\\{(0,0)\\}\) is not affine. \([\text {Hint}: \text { Show that }((X) \cong h[x, 1] \text { and use }(3.5)\) See (III, Ex. 4.3 for another proof.

Any nonempty open subset of an irreducible topological space is dense and irreducible. If \(Y\) is a subset of a topological space \(X,\) which is irreducible in its induced topology, then the closure \(\bar{Y}\) is also irreducible.

Let \(P_{1}, \ldots, P_{r}, Q_{1}, \ldots, Q_{3}\) be distinct points of \(\mathbf{A}^{1} .\) If \(\mathbf{A}^{1}-\left\\{P_{1}, \ldots, P_{r}\right\\}\) is isomorphic to \(\mathbf{A}^{1}-\left\\{Q_{1}, \ldots, Q_{s}\right\\}\) show that \(r=s\). Is the converse true?

Let \(Y\) be a variety of dimension \(r\) and degree \(d>1\) in \(P^{n} .\) Let \(P \in Y\) be a nonsingular point. Define \(X\) to be the closure of the union of all lines \(P Q,\) where \(Q \in Y, Q \neq P\) (a) Show that \(X\) is a variety of dimension \(r+1\) (b) Show that \(\operatorname{deg} X

Let \(Y \subseteq \mathbf{P}^{2}\) be a nonsingular plane curve of degree \(>1\), defined by the equation \(f(x, y, z)=0 .\) Let \(X \subseteq \mathbf{A}^{3}\) be the affine variety defined by \(f\) (this is the cone over \(Y ;\) see (Ex. 2.10) ). Let \(P\) be the point \((0,0,0),\) which is the vertex of the cone. Let \(\varphi: \tilde{X} \rightarrow X\) be the blowing-up of \(X\) at \(P\) (a) Show that \(X\) has just one singular point, namely \(P\) (b) Show that \(\tilde{X}\) is nonsingular (cover it with open affines). (c) Show that \(\varphi^{-1}(P)\) is isomorphic to \(Y\)

(a) Show that the following conditions are equivalent for a topological space \(X:\) (i) \(X\) is noetherian; (ii) every nonempty family of closed subsets has a minimal element; (iii) \(X\) satisfies the ascending chain condition for open subsets; (iv) every nonempty family of open subsets has a maximal element. (b) A noetherian topological space is quasi-compact, i.e., every open cover has a finite subcover. (c) Any subset of a noetherian topological space is noetherian in its induced topology. (d) A noetherian space which is also Hausdorff must be a finite set with the discrete topology.

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 \(\mathbf{P}^{r+1}(\mathrm{E} \mathrm{x} .5 .12)\).

Let \(Y \subseteq \mathbf{P}^{n}\) be a projective variety of dimension \(r .\) Let \(f_{1}, \ldots, f_{t} \in S=\) \(k\left[x_{0}, \ldots, x_{n}\right]\) be homogeneous polynomials which generate the ideal of \(Y\). Let \(P \in Y\) be a point, with homogeneous coordinates \(P=\left(a_{0}, \ldots, a_{n}\right) .\) Show that \(P\) is nonsingular on \(Y\) if and only if the rank of the matrix \(\left\|\left(\partial f_{i} / \partial x_{j}\right)\left(a_{0}, \ldots, a_{n}\right)\right\|\) is \(n-r .\) [Hint: (a) Show that this rank is independent of the homogeneous coordinates chosen for \(P ;\) (b) pass to an open affine \(U_{i} \subseteq \mathbf{P}^{n}\) containing \(P\) and use the affine Jacobian matrix; (c) you will need Euler's lemma, which says that if \(\left.f \text { is a homogeneous polynomial of degree } d, \text { then } \sum x_{i}\left(\partial f / \partial x_{i}\right)=d \cdot f .\right]\)

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 hypersurfuce in \(\mathbf{P}^{\prime \prime}\)

Let \(f \in k[x, y, z]\) be a homogeneous polynomial, let \(Y=Z(f) \subseteq \mathbf{P}^{2}\) be the algebraic set defined by \(f,\) and suppose that for every \(P \in Y\), at least one of \((\partial f / \partial x)(P),(\partial f / \partial y)(P),(\partial f / \partial z)(P)\) is nonzero. Show that \(f\) is irreducible (and hence that \(Y \text { is a nonsingular variety). }[\text {Hint}: \text { Use (Ex. } 3.7) .]\)

The homogeneous coordinate ring of a projective variety is not invariant under isomorphism. For example, let \(X=\mathbf{P}^{1}\). and let \(Y\) be the 2 -uple embedding of \(\mathbf{P}^{1}\) in \(\mathbf{P}^{2} .\) Then \(X \cong Y\left(\mathrm{E}_{X} .3 .4\right) .\) But show that \(S(X) \not=S\left(Y^{\prime}\right)\)

Projective Closure of an Affine Varicty. 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 \(\bar{Y},\) the closure of \(Y\) in \(\mathbf{P}^{n}\), which is called the projectice closure of \(Y\) (a) Show that \(I(\bar{Y})\) is the ideal generated by \(\beta(I(Y)),\) using the notation of the proof of (2.2) (b) Let \(Y \subseteq \mathbf{A}^{3}\) be the twisted cubic of (Ex. 1.2 ). Its projective closure \(\bar{Y} \subseteq \mathbf{P}^{3}\) is called the twisted cubic curce in \(\mathbf{P}^{3}\). Find generators for \(I(Y)\) and \(I(\bar{Y})\), 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 \(I(\bar{Y})\)

Let \(\mathfrak{a} \subseteq A=k\left[x_{1}, \ldots, x_{n}\right]\) be an ideal which can be generated by \(r\) elements. Then every irreducible component of \(Z(a)\) has dimension \(\geqslant n-r\).

For a point \(P\) on a variety \(X,\) let \(m\) be the maximal ideal of the local ring \(\mathscr{O}_{P}\) We define the Zariski tangent space \(T_{P}(X)\) of \(X\) at \(P\) to be the dual \(k\) -vector space of \(\mathrm{m} / \mathrm{m}^{2}\) (a) For any point \(P \in X, \operatorname{dim} T_{P}(X) \geqslant \operatorname{dim} X,\) with equality if and only if \(P\) is nonsingular. (b) For any morphism \(\varphi: X \rightarrow Y\), there is a natural induced \(k\) -linear map \(T_{P}(\varphi)\) \(T_{P}(X) \rightarrow T_{\varphi(P)}(Y)\) (c) If \(\varphi\) is the vertical projection of the parabola \(x=y^{2}\) onto the \(x\) -axis, show that the induced map \(T_{0}(\varphi)\) of tangent spaces at the origin is the zero map.

Let 1 be the cuspidal cubic curve \(1^{2}=\mu^{3}\) in \(\mathbf{A}^{2}\). Blow up the point \(O=(0.0).\) let \(E\) be the exceptional curve. and let \(Y\) be the strict transform of ). Show that \(E\) meets \(\tilde{Y}\) in one point. and that \(\tilde{Y} \cong \mathbf{A}^{1}\). In this case the morphism \(\varphi: \tilde{Y} \rightarrow 1\) is bijcctive and bicontinuous. but it is not an isomorphism.

The Cone Orer a Projective Variety (Fig.1). 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 $$C(Y)=\theta^{-1}(Y) \cup\\{(0, \ldots, 0)\\}$$ (a) 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 \(k\left[x_{0}, \ldots, x_{n}\right]\) (b) \(C(Y)\) is irreducible if and only if \(Y\) is. (c) \(\operatorname{dim} C(Y)=\operatorname{dim} Y+1\) Sometimes we consider the projective closure \(\overline{C(Y)}\) of \(C(Y)\) in \(P^{n+1}\). This is called the projective cone over \(Y\)

(a) If \(Y\) is any subset of a topological space \(X,\) then \(\operatorname{dim} Y \leqslant \operatorname{dim} X\) (b) If \(X\) is a topological space which is covered by a family of open subsets \(\left\\{U_{i}\right\\}\) then \(\operatorname{dim} X=\sup \operatorname{dim} U_{i}\) (c) Give an example of a topological space \(X\) and a dense open subset \(U\) with \(\operatorname{dim} U<\operatorname{dim} X\) (d) If \(Y\) is a closed subset of an irreducible finite-dimensional topological space \(X\) and if \(\operatorname{dim} Y=\operatorname{dim} X,\) then \(Y=X\) (e) Give an example of a noetherian topological space of infinite dimension.

The Elliptic Quartic Curve in \(\mathbf{P}^{3}\). Let \(Y\) be the algebraic set in \(\mathbf{P}^{3}\) defined by the equations \(x^{2}-x z-y w=0\) and \(y z-x w-z w=0 .\) Let \(P\) be the point \((x, y, z, w)=(0,0,0,1),\) and let \(\varphi\) denote the projection from \(P\) to the plane \(w=0\). Show that \(\varphi\) induces an isomorphism of \(Y-P\) with the plane cubic curve \(y^{2} z-x^{3}+x z^{2}=0\) minus the point \((1,0,-1) .\) Then show that \(Y\) is an irreducible nonsingular curve. It is called the elliptic quartic curve in \(\mathbf{P}^{3}\). since it is defined by two equations it is another example of a complete intersection (Ex. 2.17 ).

Linear Varieties in \(\mathbf{P}^{n}\). A hypersurface defined by a linear polynomial is called a hyperplane (a) Show that the following two conditions are equivalent for a variety \(Y\) in \(\mathbf{P}^{\prime \prime}:\) (i) \(I(Y)\) can be generated by linear polynomials. (ii) \(Y\) can be written as an intersection of hyperplanes. In this case we say that \(Y\) is a linear rariety in \(\mathbf{P}^{\prime \prime}\) (b) If \(Y\) is a linear variety of dimension \(r\) in \(\mathbf{P}^{\prime \prime}\), show that \(I\) ( \(Y\) ) is minimally generated by \(n-r\) linear polynomials. (c) Let \(Y, Z\) be linear varieties in \(\mathbf{P}^{n},\) with \(\operatorname{dim} Y=r, \operatorname{dim} Z=s\). 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 \(\geqslant r+s-n .\) (Think of \(\mathbf{A}^{n+1}\) as a vector spacc over \(k\) and work with its subspaces.

Let \(Y \subseteq \mathbf{A}^{3}\) be the curve given parametrically by \(x=t^{3}, y=t^{4}, z=t^{5} .\) Show that \(I(Y)\) is a prime ideal of height 2 in \(k[x, y, z]\) which cannot be generated by 2 elements. We say \(Y\) is not a local complete intersection - cf. (Ex. 2.17 ).

Quadric Hypersurfaces. Assume char \(k \neq 2,\) and let \(f\) be a homogeneous polynomial of degree 2 in \(x_{0}, \ldots, x_{n}\) (a) Show that after a suitable linear change of variables, \(f\) can be brought into the form \(f=x_{0}^{2}+\ldots+x_{r}^{2}\) for some \(0 \leqslant r \leqslant n\) (b) Show that \(f\) is irreducible if and only if \(r \geqslant 2\) (c) Assume \(r \geqslant 2\), and let \(Q\) be the quadric hypersurface in \(\mathbf{P}^{n}\) defined by \(f\). Show that the singular locus \(Z=\operatorname{sing} Q\) of \(Q\) is a linear variety (Ex. 2.11 ) of dimen\(\operatorname{sion} n-r-1 .\) In particular, \(Q\) is nonsingular if and only if \(r=n\) (d) In case \(r

The d-Uple Embedding. For given \(n, d>0,\) let \(M_{0}, M_{1}, \ldots, M_{\mathrm{Y}}\) be all the monomials of degree \(d\) in the \(n+1\) variables \(x_{0}, \ldots, x_{n},\) where \(N=\left(\begin{array}{c}n+d \\\ n\end{array}\right)-1 . \mathrm{Wc}\) define a mapping \(\rho_{d}: \mathbf{P}^{n} \rightarrow \mathbf{P}^{\prime}\) by sending the point \(P=\left(a_{0}, \ldots, a_{n}\right)\) to the point \(\rho_{d}(P)=\left(M_{0}(a), \ldots, M_{\mathrm{v}}(a)\right)\) obtained by substituting the \(a_{i}\) 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 cmbedding of \(\mathbf{P}^{1}\) in \(\mathbf{P}^{2}\) is a conic. (a) 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 \(M_{t},\) and let a be the kernel of \(0 .\) Then a is a homogencous prime ideal, and so \(Z(a)\) is a projective variety in \(\mathbf{P}^{\prime}\) (b) Show that the image of \(\rho_{d}\) is exactly \(Z\) (a). (One inclusion is casy. The other will require some calculation.) (c) Now show that \(\rho_{d}\) is a homeomorphism of \(\mathbf{P}^{n}\) onto the projective variety \(Z\) (a). (d) Show that the twisted cubic curve in \(\mathbf{P}^{3}\) (Ex. 2.9 ) is equal to the 3 -uple embed\(\operatorname{ding}\) of \(\mathbf{P}^{1}\) in \(\mathbf{P}^{3},\) for suitable choice of coordinates.

Give an example of an irreducible polynomial \(f \in \mathbf{R}[x, y],\) whose zero set \(Z(f)\) in \(\mathbf{A}_{\mathbf{R}}^{2}\) is not irreducible (cf. 1.4 .2 ).

It is a fact that any regular local ring is an integrally closed domain (Matsumura \([2, \text { Th. } 36, \text { p. } 121]\) ). Thus we see from (5.3) that any variety has a nonempty open subset of normal points (Ex. 3.17). In this exercise, show directly (without using (5.3) ) that the set of nonnormal points of a variety is a proper closed subset (you will need the finiteness of integral closure: see (3.9A)).

Analytically Isomorphic singularities. (a) If \(P \in Y\) and \(Q \in Z\) are analytically isomorphic plane curve singularities, show that the multiplicities \(\mu_{P}(Y)\) and \(\mu_{Q}(Z)\) are the same (Ex. 5.3) (b) Generalize the example in the text \((5.6 .3)\) to show that if \(f=f_{r}+f_{r+1}+\ldots \epsilon\) \(k[[x, y]],\) and if the leading form \(f_{r}\) of \(f\) factors as \(f_{r}=g_{s} h_{t},\) where \(g_{s}, h_{t}\) are homogeneous of degrees \(s\) and \(t\) respectively, and have no common linear factor, then there are formal power series $$\begin{array}{l} g=g_{s}+g_{s+1}+\ldots \\\h=h_{t}+h_{t+1}+\ldots\end{array}$$ \(\operatorname{in} k[[x, y]]\) such that \(f=g h\) (c) Let \(Y\) be defined by the equation \(f(x, y)=0\) in \(\mathbf{A}^{2},\) and let \(P=(0,0)\) be a point of multiplicity \(r\) on \(Y\), so that when \(f\) is expanded as a polynomial in \(x\) and \(y\) we have \(f=f_{r}+\) higher terms. We say that \(P\) is an ordinary \(r\) -fold point if \(f_{r}\) is a product of \(r d i s t i n c t\) linear factors. Show that any two ordinary double points are analytically isomorphic. Ditto for ordinary triple points. But show that there is a one-parameter family of mutually nonisomorphic ordinary 4-fold points. (d) Assume char \(k \neq 2 .\) Show that any double point of a plane curve is analytically isomorphic to the singularity at (0,0) of the curve \(y^{2}=x^{r}\), for a uniquely determined \(r \geqslant 2 .\) If \(r=2\) it is a node (Ex. 5.6). If \(r=3\) we call it a cusp; if \(r=4\) a tacnode. See (V, 3.9.5) for further discussion.

Products of Affine Varieties. Let \(X \subseteq \mathbf{A}^{n}\) and \(Y \subseteq \mathbf{A}^{m}\) be affine varieties. (a) Show that \(X \times Y \subseteq \mathbf{A}^{n+m}\) with its induced topology is irreducible. [Hint: Suppose that \(X \times Y\) is a union of two closed subsets \(Z_{1} \cup Z_{2}\). Let \(X_{1}=\) \(\left\\{x \in X | x \times Y \subseteq Z_{1}\right\\}, i=1,2 .\) Show that \(X=X_{1} \cup X_{2}\) and \(X_{1}, X_{2}\) are closed. Then \(\left.X=X_{1} \text { or } X_{2} \text { so } X \times Y=Z_{1} \text { or } Z_{2} .\right]\) The affine variety \(X \times Y\) is called the product of \(X\) and \(Y\). Note that its topology is in general not equal to the product topology (Ex. 1.4 ). (d) Show that \(\operatorname{dim} X \times Y=\operatorname{dim} X+\operatorname{dim} Y\) (b) Show that \(A(X \times Y) \cong A(X) \otimes_{k} A(Y)\) (c) Show that \(X \times Y\) is a product in the category of varieties, i.e., show (i) the projections \(X \times Y \rightarrow X\) and \(X \times Y \rightarrow Y\) are morphisms, and (ii) given a variety \(Z,\) and the morphisms \(Z \rightarrow X, Z \rightarrow Y,\) there is a unique morphism \(Z \rightarrow X \times Y\) making a commutative diagram

Families of Plane Curves. A homogeneous polynomial \(f\) of degree \(d\) in three variables \(x, y, z\) has \(\left(\begin{array}{c}d+2 \\ 2\end{array}\right)\) coefficients. Let these coefficients represent a point in \(\mathbf{P}^{N},\) where \(N=\left(\begin{array}{c}d+2 \\ 2\end{array}\right)-1=\frac{1}{2} d(d+3).\) (a) Show that this gives a correspondence between points of \(\mathbf{P}^{N}\) and algebraic sets in \(\mathbf{P}^{2}\) which can be defined by an equation of degree \(d\). The correspondence is \(1-1\) except in some cases where \(f\) has a multiple factor. (b) Show under this correspondence that the (irreducible) nonsingular curves of degree \(d\) correspond \(1-1\) to the points of a nonempty Zariski-open subset of \(\mathbf{P}^{N} .[\text { Hints: }(1) \text { Use elimination theory }(5.7 \mathrm{A})\) applied to the homogeneous polynomials \(\partial f / \partial x_{0}, \ldots, \partial f / \partial x_{n} ;(2)\) use the previous (Ex. 5.5, 5.8, 5.9) above.

The Quadric Surface in \(\mathbf{P}^{3}\) (Fig. 2 ). Consider the surface \(Q\) (a surfuce is a variety of dimension 2 ) in \(\mathbf{P}^{3}\) defined by the equation \(x y-z w=0\) (a) 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. (b) Show that \(Q\) contains two families of lines (a line is a linear varicty of dimension \(11: L_{1} ; \ldots . M_{t}^{\prime} .\) 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 \(M_{t} \neq M_{u} \cdot M_{t} \cap M_{u}=\varnothing \cdot\) and for all \(t . u\) \(L_{t} \cap M_{u}=\) one point (c) 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).

Products of Quasi-Projective Varicties. Use the Segre embedding (Ex. 2.14) to identify \(\mathbf{P}^{n} \times \mathbf{P}^{m}\) with its image and hence give it a structure of projective variety. Now for any two quasi-projective varieties \(X \subseteq \mathbf{P}^{n}\) and \(Y \subseteq \mathbf{P}^{m}\), consider \(X \times Y \subseteq \mathbf{P}^{n} \times \mathbf{P}^{m}\) (a) Show that \(X \times Y\) is a quasi-projective variety. (b) If \(X, Y\) are both projective, show that \(X \times Y\) is projective. (c) Show that \(X \times Y\) is a product in the category of varieties.