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I.3: Morphisms

3.1 #to_work

• Show that any conic in \(\mathbf{A}^2\) is isomorphic either to \(\mathbf{A}^1\) or \(\mathbf{A}^1\setminus\left\{{0}\right\}\) (cf. Ex.1.1).
• Show that \(\mathbf{A}^1\) is not isomorphic to any proper open subset of itself. ¹
• Any conic in \({\mathbb{P}}^2\) is isomorphic to \({\mathbb{P}}^1\).
• 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 \({\mathbb{P}}^2\).
• If an affine variety is isomorphic to a projective variety, then it consists of only one point.

3.2 #to_work

A morphism whose underlying map on the topological spaces is a homeomorphism need not be an isomorphism.

• For example, let \(\varphi: \mathbf{A}^1 \rightarrow \mathbf{A}^2\) be defined by \(t \mapsto\left(t^2, t^3\right)\). Show that \(\varphi\) defines a bijective bicontinuous morphism of \(\mathbf{A}^1\) onto the curve \(y^2=x^3\), but that \(\varphi\) is not an isomorphism.
• For another example. let the characteristic of the base field \(k\) be \(p>0\), and define a map \(\rho: \mathbf{A}^1 \rightarrow \mathbf{A}^1\) by \(t \mapsto t^p\). Show that \(\varphi\) is bijective and bicontinuous but not an isomorphism. This is called the Frobenius morphism.

3.3 #to_work

• Let \(\varphi: X \rightarrow Y\) be a morphism. Then for each \(P \in X, \varphi\) induces a homomorphism of local rings \(\varphi_P^*: {\mathcal{O}}_{\phi(P), Y} \rightarrow {\mathcal{O}}_{P, Y}\).
• 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 \(P \in X\).
• Show that if \(\varphi(X)\) is dense in \(Y\), then the map \(\rho_P^*\) is injective for all \(P \in X\).

3.4 #to_work

Show that the \(d{\hbox{-}}\)uple embedding of \({\mathbb{P}}^n(\mathrm{Ex} .2 .12)\) is an isomorphism onto its image.

3.5 #to_work

By abuse of language, we will say that a variety “is affine” if it is isomorphic to an affine variety. If \(H \subseteq {\mathbb{P}}^n\) is any hypersurface. show that \({\mathbb{P}}^n-H\) is affine. ²

3.6 There are quasi-affine varieties which are not affine. #to_work

For example, show that \(\mathrm{I}=\mathbf{A}^2\setminus\left\{{ (0, 0) }\right\}\) is not affine. ³

3.7 #to_work

• Show that any two curves in \({\mathbb{P}}^2\) have a nonempty intersection.
• More generally, show that if \(Y \subseteq {\mathbb{P}}^n\) is a projective variety of dimension \(\geqslant 1\). and if \(H\) is a hypersurface. then \(Y \cap H \neq \varnothing\). ⁴

3.8 #to_work

Let \(H_1\) and \(H\), be the hyperplanes in \({\mathbb{P}}^n\) defined by \(x_1=0\) and \(x_{,}=0\), with \(i \neq j\). Show that any regular function on \({\mathbb{P}}^n-\left(H_1 \cap H_1\right)\) is constant. ⁵

3.9 #to_work

The homogeneous coordinate ring of a projective variety is not invariant under isomorphism. For example, let \(X={\mathbb{P}}^1\). and let \(Y\) be the 2-uple embedding of \({\mathbb{P}}^1\) in \({\mathbb{P}}^2\). Then \(X \cong Y(\) Ex. 3.4). But show that \(S(X) \equiv S(Y)\).

3.10 Subvarieties. #to_work

A subset of a topological space is locally closed if it is an open subset of its closure. or. equivalently. if it is the intersection of an open set with a closed set.

If \(X\) is a quasi-affine or quasi-projective variety and \(Y\) is an irreducible locally closed subset. then \(I\) is also a quasi-affine (respectively, quasi-projective) variety by virtue of being a locally closed subset of the same affine or projective space. We call this the induced structure on Y. and we call \(Y\) a subvariety of \(X\).

Now let \(\varphi: X \rightarrow Y\) he a morphism. let \(X^{\prime} \subseteq X\) and \(Y^{\prime} \subseteq Y\) be irreducible locally closed subsets such that \(\varphi\left(X^{\prime}\right) \subseteq Y^{\prime}\). Show that \(\left.\varphi\right|_{X}: X^{\prime } \rightarrow Y^{\prime}\) is a morphism.

3.11 #to_work

Let \(X\) be any variety and let \(P \in X\). Show there is a 1-1 correspondence between the prime ideals of the local ring \({\mathcal{O}}_P\) and the closed subvarieties of \(X\) containing \(P\).

3.12 #to_work

If \(P\) is a point on a variety \(X\), then \(\operatorname{dim} {\mathcal{O}}_P =\operatorname{dim} X\). ⁶

3.13 The Local Ring of a Subvariety #to_work

Let \(Y \subseteq X\) be a subvariety. Let \({\mathcal{O}}_{Y,X}\) be the set of equivalence classes \(\langle L, f\rangle\) where \(L \subseteq X\) is open. \(L \cap Y \neq \varnothing\), and \(f\) is a regular function on \(L\). We say \(\langle L , f\rangle\) is equivalent to \(\left\langle{V, g}\right\rangle\) if \(f=g\) on \(U \cap V\).

Show that \({\mathcal{O}}_{Y , X}\) is a local ring, with residue field \(K(Y)\) and \(\operatorname{dimension}=\operatorname{dim} \mathrm{X}-\) \(\operatorname{dim} Y\). It is the local ring of \(Y\) on \(X\). Note if \(Y=P\) is a point we get \({\mathcal{O}}_P\). and if \(Y=X\) we get \(K(X)\). Note also that if \(Y\) is not a point, then \(K(Y)\) is not algebraically closed, so in this way we get local rings whose residue fields are not algebraically closed.

3.14 Projection from a Point. #to_work

Let \({\mathbb{P}}^n\) be a hyperplane in \({\mathbb{P}}^{n+1}\) and let \(P \in {\mathbb{P}}^{n+1}-{\mathbb{P}}^n\). Define a mapping \(\varphi: {\mathbb{P}}^{n+1}\setminus\left\{{ P }\right\}\to {\mathbb{P}}^n\) by \(\varphi(Q)=\) the intersection of the unique line containing \(P\) and \(Q\) with \({\mathbb{P}}^n\).

• Show that \(\varphi\) is a morphism.
• Let \(Y \subseteq {\mathbb{P}}^3\) be the twisted cubic curve which is the image of the 3-uple embedding of \({\mathbb{P}}^1\) (Ex. 2.12). If \(t,u\) are the homogeneous coordinates on \({\mathbb{P}}^1\). we say that \(Y\) is the curve given parametrically by \((x, y, z, w)=\left(t^3, t^2 u, t u^2, u^3\right)\). Let \(P=(0,0,1,0)\), and let \({\mathbb{P}}^2\) be the hyperplane \(z=0\). Show that the projection of \(Y\) from \(P\) is a cuspidal cubic curve in the plane, and find its equation.

3.15 Products of Affine Varieties. #to_work

Let \(X \subseteq \mathbf{A}^n\) and \(Y \subseteq \mathbf{A}^m\) be affine varieties.

• Show that \(X \times Y \subseteq \mathbf{A}^{n+m}\) with its induced topology is irreducible. ⁷ 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).
• Show that \(A(X \times Y) \cong A(X) \otimes_k A(Y)\).
• Show that \(X \times Y\) is a product in the category of varieties, i.e., show
  • the projections \(X \times Y \rightarrow X\) and \(X \times Y \rightarrow Y\) are morphisms, and
  • 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

Link to Diagram

• Show that \(\operatorname{dim} X \times Y=\operatorname{dim} X+\operatorname{dim} Y\).

3.16 Products of Quasi-Projective Varieties. #to_work

Use the Segre embedding (Ex. 2.14) to identify \({\mathbb{P}}^n \times {\mathbb{P}}^m\) with its image and hence give it a structure of projective varieties. Now for any two quasi-projective varieties \(X \subseteq {\mathbb{P}}^n\) and \(Y \subseteq {\mathbb{P}}^m\), consider \(X \times Y \subseteq {\mathbb{P}}^n \times {\mathbb{P}}^m\).

• Show that \(X \times Y\) is a quasi-projective variety.
• If \(X, Y\) are both projective, show that \(X \times Y\) is projective.
• Show that \(X \times Y\) is a product in the category of varieties.

3.17 Normal Varieties. #to_work

A variety \(Y\) is normal at a point \(P \in Y\) if \({\mathcal{O}}_P\) is an integrally closed ring. \(Y\) is normal if it is normal at every point.

• Show that every conic in \({\mathbb{P}}^2\) is normal.
• Show that the quadric surfaces \(Q_1, Q_2\) in \(\mathrm{P}^3\) given by equations \(Q_1: x y=zw\); \(Q_2: xy=z^2\) are normal. (cf. (II. Ex. 6.4) for the latter.)
• Show that the cuspidal cubic \(y^2=x^3\) in \(\mathbf{A}^2\) is not normal.
• If \(Y\) is affine, then \(Y\) is normal \(\Leftrightarrow A(Y)\) is integrally closed.
• Let \(Y\) be an affine variety. Show that there is a normal affine variety \(\tilde{Y}\), and a morphism \(\pi: \tilde{Y} \rightarrow Y\), with the property that whenever \(Z\) is a normal variety, and \(\varphi: Z \rightarrow Y\) is a dominant morphism (i.e., \(\varphi(Z)\) is dense in \(Y\)), then there is a unique morphism \(\theta: Z \rightarrow \tilde{Y}\) such that \(\varphi=\pi \quad \theta\). \(\tilde{Y}\) is called the normalization of \(Y\). You will need \((3.9 \mathrm{~A})\) above.

3.18 Projectively Normal Varieties. #to_work

A projective variety \(Y \subseteq \mathrm{P}^n\) is projectively normal (with respect to the given embedding) if its homogeneous coordinate ring \(S\left(Y \right)\) is integrally closed.

• If \(Y\) is projectively normal, then \(Y\) is normal.
• There are normal varieties in projective space which are not projectively normal. For example, let \(Y\) be the twisted quartic curve in \({\mathbb{P}}^3\) given parametrically by \((x, y: z, w)=\left(t^4, t^3 u, t u^3, u^4\right)\). Then \(Y\) is normal but not projectively normal. See (III, Ex. 5.6) for more examples.
• Show that the twisted quartic curve \(Y\) above is isomorphic to \({\mathbb{P}}^1\). which is projectively normal. Thus projective normality depends on the embedding.

3.19 Automorphisms of \(\mathbf{A}^n\). #to_work

Let \(\varphi: \mathbf{A}^n \rightarrow \mathbf{A}^n\) be a morphism of \(\mathbf{A}^n\) to \(\mathbf{A}^n\) given by \(n\) polynomials \(f_1 \ldots . f_n\) of \(n\) variables \(x_1, \ldots x_n\). Let \(J=\operatorname{det}\left[ {\frac{\partial f_i}{\partial x_j}\,} \right]\) be the Jacobian polynomial of \(\varphi\).

• If \(\varphi\) is an isomorphism (in which case we call \(\varphi\) an automorphism of \(\mathbf{A}^n\) ) show that \(J\) is a nonzero constant polynomial.
• ** The converse of 1.is an unsolved problem, even for \(n=2\). See, for example, Vitushkin.

3.20 #to_work

Let \(Y\) be a variety of dimension \(\geqslant 2\), and let \(P \in Y\) be a normal point. Let \(f\) be a regular function on \(Y-P\).

• Show that \(f\) extends to a regular function on \(Y\).
• Show this would be false for \(\operatorname{dim} Y=1\). See (III. Ex. 3.5) for generalization.

3.21. Group Varieties. #to_work

A group variety consists of a variety Y together with a morphism \(\mu: Y \times Y \rightarrow Y\). such that the set of points of \(Y\) with the operation given by \(\mu\) is a group. and such that the inverse map \(y^{-y^{-1}}\) is also a morphism of \(Y \rightarrow Y\).

• The additive group \(\mathbf{G}_a\) is given by the variety \(\mathbf{A}^1\) and the morphism \(\mu: \mathbf{A}^2 \rightarrow \mathbf{A}^1\) defined by \(\mu(a, b) = a+b\). Show it is a group variety.
• The multiplicative croup \(\mathbf{G}_m\) is given by the variety \(\mathbf{A}^1\setminus\left\{{0}\right\}\), and the morphism \(\mu(a, b) = ab\). Show \(|\) in a group variety.
• If \(G\) is a group variety, and \(X\) is any variety. show that the set \(\operatorname{Hom}(X, G)\) has a natural group structure.
• For any variety \(X\), show that \(\operatorname{Hom}\left(X, \mathbf{G}_a\right)\) is isomorphic to (’ (X) as a group under addition.
• For any variety \(X\), show that \(\operatorname{Hom}\left(X, \mathbf{G}_m\right)\) is isomorphic to the group of units in \({\mathcal{O}}(X)\), under multiplication.