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The Geometry of Schemes

David Eisenbud, Joe Harris

Chapter 6

Schemes and Functors - all with Video Answers

Educators

Chapter Questions

Problem 1

Let $\mathscr{C}$ be a category and let $X, X^{\prime}$ be objects of $\mathscr{C}$.
(a) If $F$ is any contravariant functor from $\mathscr{C}$ to the category of sets, the natural transformations from $\operatorname{Mor}(-, X)$ to $F$ are in natural correspondence with the elements of $F(X)$.
(b) If the functors $\operatorname{Mor}(-, X)$ and $\operatorname{Mor}\left(-, X^{\prime}\right)$ from $\mathscr{C}$ to the category of sets are isomorphic, then $X \simeq X^{\prime}$. More generally, the maps of functors from $\operatorname{Mor}(-, X)$ to $\operatorname{Mor}\left(-, X^{\prime}\right)$ are the same as maps from $X$ to $X^{\prime}$; that is, the functor
$$
h: \mathscr{C} \rightarrow \operatorname{Fun}\left(\mathscr{C}^{\circ},(\text { sets })\right)
$$

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Problem 2

If $R$ is a commutative ring, a scheme over $R$ is determined by the restriction of its functor of points to affine schemes over $R$; in fact
$$
h:(R \text {-schemes }) \rightarrow \text { Fun }((R \text {-algebras }),(\text { sets }))
$$
is an equivalence of the category of $R$-schemes with a full subcategory of the category of functors.

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Problem 3

Suppose that $X$ is (like virtually all schemes of interest to us) locally Noetherian - that is, covered by spectra of Noetherian rings. Prove that $X$ is determined by the restriction of $h_X$ to the category of Noetherian rings.

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Problem 4

If $A, B$, and $C$ are functors from some category $\mathscr{C}$ to the category of sets and if $f: A \rightarrow C$ and $g: B \rightarrow C$ are morphisms of functors, the fibered product $A \times_C B$ is the functor from $\mathscr{C}$ to (sets) defined by setting, for any object $Z$ of $\mathscr{C}$,
$$
\left(A \times_C B\right)(Z)=\{(a, b) \in A(Z) \times B(Z) \mid f(a)=f(b) \text { in } C(Z)\},
$$
and defined on morphisms of $\mathscr{C}$ in the obvious way.

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Problem 5

A subfunctor $\alpha: G \rightarrow F$ in Fun((rings), (sets)) is an open subfunctor if, for each map $\psi: h_{\mathrm{Spec} R} \rightarrow F$ from the functor represented by an affine scheme Spec $R$ (that is, each $\psi \in F(R)$ ), the fibered product of functors yields a map $G_\psi \rightarrow h_{\mathrm{Spec} R}$ isomorphic to the injection from the functor represented by some open subcheme of $\operatorname{Spec} R$.

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Problem 6

Let $X=\operatorname{Spec} R$ be an affine scheme. Show that the open subfunctors of $h_X$ are exactly the functors of the form
$$
F(T)=\left\{\varphi \in h_X(T) \mid \varphi^*(I) T=T\right\},
$$
for some ideal $I \subset R$.

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Problem 7

Let $X$ be a scheme over the field $K$. Define a functor $F:(\text { schemes } / K)^{\circ} \rightarrow$ (sets) as follows: for each $K$-scheme $Y$, let $F(Y)$ be the set of closed subschemes $Z \subset X \times_K Y$ flat over $Y$ and such that all the fibers of $Z$ over closed points of $Y$ are subschemes of degree 2 of $X$. Let $G$ be the subfunctor of $F$ obtained by adding the requirement that the fibers of $Z$ over closed points of $Y$ are reduced. Show that $G \subset F$ is open.

To define closed functors, we proceed similarly. A subfunctor $\alpha: G \rightarrow F$ in Fun((rings), (sets)) is closed if for each map $\psi: h_{\mathrm{Spec} R} \rightarrow F$ the fibered product of $\psi$ and $\alpha$ is a subfunctor of $h_{\mathrm{Spec} R}$ isomorphic to the functor represented by a closed subcheme of Spec $R$.

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Problem 8

Let $X=\operatorname{Spec} R$ be an affine scheme. Show that the open and closed subfunctors of $h_{\mathrm{Spec} R}$ are precisely those represented by open and closed subschemes of Spec $R$. (The same is true, and only a little harder, for arbitrary schemes.)

As usual, a little caution is necessary when using these notions. For example:

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Problem 9

Suppose that $F$ is a functor from a category $\mathscr{C}$ to the category of sets, and let $G$ be a subfunctor of $F$. Show by example that the association $C \mapsto F(C) \backslash G(C)$ may not define a functor. Suppose now that $\mathscr{C}$ is the category of rings and $F$ is represented by a scheme $X$ while $G$ is represented by a closed subscheme $Y$ of $X$, and $H$ is the functor reprsented by the open subscheme $X \backslash Y$. Can you describe $H$ in terms of $G$ and $F$ ?

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Problem 10

Let $X$ be a scheme over the field $K$. Define the functor $F:(\text { scheme } / K)^{\circ} \rightarrow$ (sets) as in Exercise VI-7, and define a subfunctor $H$ of $F$ by letting $H(Y)$, for each $K$-scheme $Y$, be the set of closed subschemes $Z \subset X \times Y$ flat over $Y$ and such that all the fibers of $Z$ over closed points of $Y$ are subschemes of degree 2 of $X$ supported at a single point of $X$. Show that $H$ is not in general a closed subfunctor of $F$.

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Problem 11

Let $\left\{G_i \rightarrow F\right\}$ be a collection of open subfunctors of a functor $F:$ (schemes) $\rightarrow$ (sets). Show that this is an open covering if and only if $F(\operatorname{Spec} K)=\bigcup G_i(\operatorname{Spec} K)$ for all fields $K$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 12

Let $X=$ Spec $\mathbb{C}$, considered as an abstract scheme, that is, a scheme over $\mathbb{Z}$. Describe the set $h_X(\operatorname{Spec} \mathbb{C})$ of all $\mathbb{C}$-valued points of Spec $\mathbb{C}$.

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Problem 13

Verify that these maps make $T_p F$ into a $K$-vector space, and that this is the old vector space structure in the case where $F$ is a representable functor.

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Problem 14

A functor $F:$ (rings) $\rightarrow$ (sets) is of the form $h_Y$ for some scheme $Y$ if and only if
(1) $F$ is a sheaf in the Zariski topology, and
(2) there exist rings $R_i$ and open subfunctors
$$
\alpha_i: h_{R_i} \rightarrow F
$$
such that, for every field $K, F(K)$ is the union of the images of $h_{R_i}(K)$ under the maps $\alpha_i$.

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Problem 15

(a) Show that if $f: A \rightarrow C$ and $g: B \rightarrow C$ are morphisms of functors all of which are sheaves in the Zariski topology, the fibered product $A \times_C B$ is a sheaf in the Zariski topology.
(b) Use the open covering of the fibered product suggested in Chapter I and the above theorem to prove the existence of fibered products in the category of schemes.
The next example gives a different way of looking at maps to projective spaces. Theorem III-37 may be translated immediately into our new language:

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Problem 16

If $Y=\mathbb{P}_{\mathrm{Z}}^n$, then
$$
\begin{aligned}
h_Y(X) & =\left\{\begin{array}{l}
\text { locally free subsheaves } \mathscr{F} \subset \mathscr{O}_X^{n+1} \\
\text { that locally are summands of rank } n
\end{array}\right\} \\
& =\frac{\left\{\text { invertible sheaves } P \text { on } X \text { with an epimorphism } \mathscr{O}_X^{n+1} \rightarrow P\right\}}{\{\text { isomorphisms }\}},
\end{aligned}
$$
where isomorphism is defined as in Corollary III-42.

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04:47

Problem 17

Check that these definitions are independent of all the choices made.

Finally, Theorem VI-14 can also be used to prove the existence of the Grassmannian scheme from its functorial description.

Daniel Jaimes
Daniel Jaimes
Numerade Educator

Problem 18

For $0<k<n$, let
$$
g=g(k, n): \text { (rings) } \rightarrow \text { (sets) }
$$
be the Grassmannian functor, that is, the functor given by
$$
g(T)=\left\{\text { rank } k \text { direct summands of } T^n\right\} .
$$

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Problem 19

Show that this definition of the Grassmannian coincides with the one given in Section III.2.7.

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Problem 20

For a field $K$, give an analogous definition of the Grassmannian $G_K(k, n)$, and show that it coincides with the product $G_{\mathrm{Z}}(k, n) \times$ Spec $K$.

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Problem 21

The Hilbert functor $h_P$-called the "functor of flat families of schemes in $\mathbb{P}_{\mathrm{Z}}^n$ with Hilbert polynomial $P^{\prime \prime}$ - is the functor
$$
h_P: \text { (schemes) }{ }^{\circ} \rightarrow \text { (sets) }
$$
that associates to any $B$ the set of subschemes $\mathscr{X} \subset \mathbb{P}_B^n$ flat over $B$ whose fibers over points of $B$ have Hilbert polynomial $P$.

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Problem 22

There exists a scheme $\mathscr{H}_P$ whose functor of points is the functor $h_P$.

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Problem 23

Let $P(m)=m+1$ be the Hilbert polynomial of a line. Show that the Hilbert scheme of subschemes of $\mathbb{P}_Z^3$ with Hilbert polynomial $P$ is the Grassmannian introduced in Exercise VI-20.

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Problem 24

Let $P(m)=2 m+1$ be the Hilbert polynomial of a conic curve. Show that the Hilbert scheme $\mathscr{H}_P$ of subschemes of $\mathbb{P}_Z^2$ with Hilbert polynomial $P$ is $\mathbb{P}_Z^5$.

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Problem 25

Let $P$ be the constant polynomial 2. Show that the Hilbert scheme $\mathscr{H}_P$ parametrizing subschemes of $\mathbb{P}_Z^2$ with Hilbert polynomial $P$ may be obtained by blowing up the product $\mathbb{P}_Z^2 \times \mathbb{P}_Z^2$ along the diagonal and then taking the quotient by the involution exchanging factors.

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Problem 26

Show that $h_{P, X}$ is a closed subfunctor of $h_P$.
It follows that there exists a closed subscheme $\mathscr{H}_{P, X} \subset \mathscr{H}_P$ whose functor of points is $h_{P, X}$; this scheme is what we call the Hilbert scheme of subschemes of $X$ with Hilbert polynomial $P$.

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Problem 27

Fix two integers $g, h \geq 2$. Show that there is a number $N(g, h)$ such that for any nonsingular curves $C$ and $C^{\prime}$ of genera $g$ and $h$ respectively, the number of maps from $C$ to $C^{\prime}$ is less than $N(g, h)$.

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Problem 28

Let $X \subset \mathbb{P}_S^m$ and $Y \subset \mathbb{P}_T^n$ be schemes flat over the $K$-schemes $S$ and $T$, and suppose that for any pair of closed points $s \in$ $S$ and $t \in T$ the number $n_d(s, t)$ of morphisms of degree $d$ between the corresponding fibers $X_s$ and $Y_t$ is finite. Show that, for fixed $d, n_d(s, t)$ is bounded as $s$ and $t$ vary.

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Problem 29

Given a closed subscheme $X$ of a scheme $Y$, the space of first-order deformations of $X$ in $Y$ is the space of global sections of its normal sheaf $\mathscr{N}_{X / Y}$.

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Problem 30

The dimension of any irreducible component $\Sigma$ of the Hilbert scheme is at most the dimension of the space of sections of the normal sheaf of any scheme $X$ with $[X] \in \Sigma$.

In fact, this a priori estimate for the dimension of a component of the Hilbert scheme gives the right answer more often than not, especially when applied to a general point $[X]$ of a component of $\mathscr{H}_P$. The following exercises give examples of this.

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Problem 31

Let $\Sigma$ be the component of the Hilbert scheme whose general member is a complete intersection $X \subset \mathbb{P}_K^n$ of $k$ hypersurfaces of degree $d$. Calculate the dimension of the space of global sections of $\mathscr{N}_X$ and show that this is equal to the dimension of $\Sigma$.

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Problem 32

Generalize the preceding exercise to the case of the component of the Hilbert scheme whose general member is a complete intersection $X \subset \mathbb{P}_K^n$ of hypersurfaces of degrees $d_1, \ldots, d_k$. (This can get complicated; you may want to stick to the case $k=2$, which is enough to see how it goes.)

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Problem 33

Let $P(m)$ be the polynomial $3 m+1$ and let $\Sigma$ be the component of the Hilbert scheme $\mathscr{H}_P$ of subschemes of $\mathbb{P}_K^3$ whose general member is a twisted cubic curve $C$. Show that the dimension of $\Sigma$ is 12, and that this is equal to the dimension of the space of sections of $\mathscr{N}_C$.

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Problem 34

By way of warning, the component $\Sigma$ of the preceding exercise is not the only component of the Hilbert scheme: there is another component $\Sigma^{\prime}$ whose general member is the (disjoint) union of a plane cubic curve and a point. These two intersect in the locus of schemes $C \subset \mathbb{P}_K^3$ such that $C$ is the union of a plane curve $C_0$ and the double point (that is, the scheme given by the square of the maximal ideal of a point) supported at a singular point of $C_0$. At a point of their intersection $\mathscr{H}_P$ is singular, and its tangent space will be strictly larger than the dimension of either $\Sigma$ or $\Sigma^{\prime}$. Verify this.

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Problem 35

Show that $\Sigma$ is a constructible subset of $\mathscr{H}$ and that its closure $\bar{\Sigma}$ in $\mathscr{H}$ has dimension 56 .

Not all curves of degree 14 and genus 24 in $\mathbb{P}_K^3$ have to lie on cubic surfaces. Thus, it is not a priori clear that the subvariety $\bar{\Sigma} \subset \mathscr{H}$ is an irreducible component of $\mathscr{H}$ : the curves $C$ parametrized by $\bar{\Sigma}$ could be specializations of other curves not lying on cubics. To see that this is not in fact the case, we make another dimension count.

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Problem 36

Let $C$ be a nonsingular, irreducible curve of degree 14 and genus 24 in $\mathbb{P}_K^3$, and assume that $C$ does not lie on a cubic surface. Show that it must lie on two quartic surfaces $T, T^{\prime}$ not having a common component, and that the residual intersection of $T$ and $T^{\prime}$ (that is, the union of the irreducible components of $T \cap T^{\prime}$ other than $C$ ) is a curve of degree 2. By analyzing what this residual intersection may look like, show that the set of such curves $C$ is a constructible subset of $\mathscr{H}$ whose closure has dimension at most 56. Deduce that the subvariety $\bar{\Sigma}$ of Exercise VI-35 is indeed an irreducible component of $\mathscr{H}$.

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Problem 37

Now let $C$ be a nonsingular curve of degree 14 and genus 24 lying on a nonsingular cubic surface $S \subset \mathbb{P}_K^3$. Using the exact sequence
$$
0 \rightarrow \mathscr{N}_{C / S} \rightarrow \mathscr{N}_{C / \mathbb{P}_K^3} \rightarrow \mathscr{N}_{S / \mathbb{P}_K^3} \otimes \mathscr{O}_C \rightarrow 0
$$
(where for any pair of schemes $X \subset Y$ we write $\mathscr{N}_{X / Y}$ for the normal sheaf $\operatorname{Hom}\left(\mathscr{I}_{X / Y}, \mathscr{O}_X\right)$ of $X$ in $Y$ ), show that the dimension of the space of sections of the normal sheaf $\mathscr{N}_{C / P_K^3}$ is 57 . Deduce that $\mathscr{H}$ is nowhere reduced along $\bar{\Sigma}$.

Finally, here is an amusing fact about the Hilbert scheme of rational normal curves, generalizing a calculation we made in Section IV.4.

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Problem 38

Let $K$ be a field. For any $r$, let $P(m)$ be the polynomial $r m+1$, and let $\Sigma$ be the open subset of the Hilbert scheme $\mathscr{H}_P$ of subschemes of $\mathbb{P}_K^r$ parametrizing rational normal curves of degree $r$; check that $\Sigma$ is irreducible of dimenion $r^2+2 r-3$. Let $\mathscr{C} \subset \Sigma \times \mathbb{P}_K^r \rightarrow \Sigma$ be the universal curve over $\Sigma$. Let $L$ be the function field of $\Sigma$, and $C_L$ the fiber of $\mathscr{C}$ over the generic point Spec $L$ of $\Sigma$. Show that $C_L \cong \mathbb{P}_L^1$ if and only if $r$ is odd.

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Problem 39

The functor $f_k(X)$ is represented $F_k(X)$ introduced in Section IV.3.

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Problem 40

Use this characterization of the tangent spaces to Fano schemes to give an example of a nonsingular hypersurface $X \subset \mathbb{P}_K^n$ such that the Fano scheme $F_1(X)$ of lines on $X$ is singular.

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