111.65 Algebraic Geometry, Final Exam, Spring 2022

These were the questions in the final exam of a course on Algebraic Geometry, in the Spring of 2022 at Columbia University.

Exercise 111.65.1 (Definitions). Provide brief definitions of the italicized concepts.

1. a scheme,

2. a quasi-coherent module on a scheme $X$,

3. a flat morphism of schemes $X \to Y$,

4. a finite morphism of schemes $X \to Y$,

5. a group scheme $G$ over a base scheme $S$,

6. a family of varieties over a base scheme $S$,

7. the degree of a closed point $x$ on a variety $X$ over the field $k$,

8. the usual logarithmic height of a point $p = (a_0 : \ldots : a_ n)$ in $\mathbf{P}^ n(\mathbf{Q})$, and

9. a $C_ i$ field.

Exercise 111.65.2 (Theorems). Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. morphisms from a scheme $X$ to the affine scheme $\mathop{\mathrm{Spec}}(A)$,

2. cohomology of a quasi-coherent module $\mathcal{F}$ on an affine scheme $X$,

3. the Picard group of $\mathbf{P}^1_ k$ where $k$ is a field,

4. the dimensions of fibres of a flat proper morphism $X \to S$ for $S$ Noetherian,

5. $\mathbf{G}_ m$-equivariant modules on a scheme $S$, and

6. Bezout's theorem on intersections (restrict to a special case if you like).

Exercise 111.65.3 (Cubic hypersurfaces). Let $F \in \mathbf{C}[T_0, \ldots , T_ n]$ be homogeneous of degree $3$. Given $3$ vectors $x, y, z \in \mathbf{C}^{n + 1}$ consider the condition

$(*)\quad F(\lambda x + \mu y + \nu z) = 0 \text{ in } \mathbf{C}[\lambda , \mu , \nu ]$

1. What is the dimension of the space of all choices of $x, y, z$?

2. How many equations on the coordinates of $x$, $y$, and $z$ is condition (*)?

3. What is the expected dimension of the space of all triples $x, y, z$ such that (*) is true?

4. What is the dimension of the space of all triples such that $x, y, z$ are linearly dependent?

5. Conclude that on a hypersurface of degree $3$ in $\mathbf{P}^ n$ we expect to find a linear subspace of dimension $2$ provided $n \geq a$ where it is up to you to find $a$.

Exercise 111.65.4 (Heights). Let $K$ be a field. Let $h_ n : \mathbf{P}^ n(K) \to \mathbf{R}$, $n \geq 0$ be a collection of functions satisfying the $2$ axioms we discussed in the lectures. Let $X$ be a projective variety over $K$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module and recall that we have constructed in the lectures an associated height function $h_\mathcal {L} : X(K) \to \mathbf{R}$. Let $\alpha : X \to X$ be an automorphism of $X$ over $K$.

1. Prove that $P \mapsto h_\mathcal {L}(\alpha (P))$ differs from the function $h_{\alpha ^*\mathcal{L}}$ by a bounded amount. (Hint: recall that if there is a morphism $\varphi : X \to \mathbf{P}^ n$ with $\mathcal{L} = \varphi ^*\mathcal{O}_{\mathbf{P}^ n}(1)$, then by construction $h_\mathcal {L}(P) = h_ n(\varphi (P))$ and play around with that. In general write $\mathcal{L}$ as a difference of two of these.)

2. Assume that $h_\mathcal {L}(P) - h_\mathcal {L}(\alpha (P))$ is unbounded on $X(K)$. Show that $h_\mathcal {N}$ with $\mathcal{N} = \mathcal{L} \otimes \alpha ^*\mathcal{L}^{\otimes -1}$ is unbounded on $X(K)$.

3. Assume $X$ is an elliptic curve and that $\mathcal{L}$ is a symmetric ample invertible module on $X$ such that $h_\mathcal {L}$ is unbounded on $X(K)$. Show that there exists an invertible module $\mathcal{N}$ of degree $0$ such that $h_\mathcal {N}$ is unbounded. (Hints: Recall that $X$ is an abelian variety of dimension $1$. Thus $h_\mathcal {L}$ is quadratic up to a constant by results in the lectures. Choose a suitable point $P_0 \in X(K)$. Let $\alpha : X \to X$ be translation by $P_0$. Consider $P \mapsto h_\mathcal {L}(P) - h_\mathcal {L}(P + P_0)$. Apply the results you proved above.)

Exercise 111.65.5 (Monomorphisms). Let $f : X \to Y$ be a monomorphism in the category of schemes: for any pair of morphisms $a, b : T \to X$ of schemes if $f \circ a = f \circ b$, then $a = b$. Show that $f$ is injective on points. Does you argument say anything else?

Exercise 111.65.6 (Fixed points). Let $k$ be an algebraically closed field.

1. If $G = \mathbf{G}_{m, k}$ show that if $G$ acts on a projective variety $X$ over $k$, then the action has a fixed point, i.e., prove there exists a point $x \in X(k)$ such that $a(g, x) = x$ for all $g \in G(k)$.

2. Same with $G = (\mathbf{G}_{m, k})^ n$ equal to the product of $n \geq 1$ copies of the multiplicative group.

3. Give an example of an action of a connected group scheme $G$ on a smooth projective variety $X$ which does not have a fixed point.

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