Varieties: Problems

Some sourced from http://www.mat.uniroma2.it/~ricerca/geomet/workshops/Zaidenbergnotes.pdf

Results

What is the Nullstellensatz? How is it proved?
What is the adjunction formula?
What is the Riemann-Roch theorem?
- What is Riemann-Roch for curves?
- What is Riemann-Roch for surfaces?
What is the Noether formula?
What is Serre duality?
What is Kodaira vanishing?
What is the Hilbert basis theorem?
What is Noether normalization?
What is Zariski’s main theorem?
What is the Stein factorization theorem?
What is the Lefschetz hyperplane section theorem?
What is the moving lemma?
What is the Néron-Severi theorem?
What is Bertini’s theorem?
What is the Cartan-Serre theorem?
What is the Nakai-Moishezon criterion?
What is the minimal model theorem for curves?
What is the minimal model program for surfaces?
What is the going down theorem?
What is Hurwitz’s theorem?

Problems

Show that any affine variety has a unique decomposition into irreducible components.
Show that the only irreducible subvarities of ${\mathbb{A}}^2_{/ {k}} $ for $k$ an infinite field are $\emptyset ,{\mathbb{A}}^2_{/ {k}} $, and irreducible plane curves $V(f)$ for $f$ an irreducible polynomial where $V(f)$ has infinite cardinality.
Show that a projective variety $X$ is irreducible iff $I(V)$ is a (homogeneous) prime ideal
Let $X \subseteq {\mathbb{A}}^n$, show that $I(X)$ is radical in ${\mathcal{O}}_{{\mathbb{A}}^n}$ and maximal iff $X$ is a point.
Show that ${\mathcal{O}}_{{\mathbb{A}}^n}$ is Noetherian.
Show that any Zariski open is dense.
Show that the Zariski topology on $X$ is never separated unless $X$ is a point.
Show that $\operatorname{Spec}k[x]$ with the Zariski topology coincides with the cofinite topology.
Show that if $X, Y$ are affine varieties, then $X\times Y$ is an affine variety with ${\mathcal{O}}_{X\times Y} \cong {\mathcal{O}}_X \otimes_k {\mathcal{O}}_Y$.
Show that any morphism ${\mathbb{A}}^1\to {\mathbb{A}}^1$ or ${\mathbb{A}}^1\setminus\left\{{0}\right\}\to {\mathbb{A}}^1\setminus\left\{{0}\right\}$ is finite.
Show that any morphism ${\mathbb{A}}^1\to{\mathbb{A}}^1\setminus\left\{{0}\right\}$ is constant.
Show that if $f:X\to Y$ is quasi-finite, then $\dim X \leq \dim Y$.
Show that if $f:X\to Y$ is finite and $\dim X = \dim Y$, then $f$ is closed and surjective.
Can a non-surjective finite morphism exist?
Let $f:X\to Y$ be a dominant morphism of affine varieties. Show that for any $y\in f(X)$, any irreducible component of the fiber $f^{-1}(y)$ is an affine variety of dimension $d\geq \dim X-\dim Y$. Show that equality holds for a Zariski-dense open subset of $Y$.
Show that if $f\in {\mathcal{O}}_X$ be nonconstant, then any irreducible component of the fiber $f^{-1}(0)$ has dimension $d = \dim X - 1$.
Let $X$ be an affine variety with $\dim X = d$. Show that if $p$ is a smooth point, $\dim {\mathbf{T}}_pX = d$, and otherwise $\dim {\mathbf{T}}_p X > d$.
Show that a point $p\in X$ is smooth iff $X \hookrightarrow{\mathbb{A}}^n$ is locally a smooth submanifold.
Show that ${\mathbf{T}}_p X \cong ({\mathfrak{m}}_p/{\mathfrak{m}}_p^2) {}^{ \vee }$.
Show that the singular points of $X$ form a proper Zariski closed subset, so smooth points form a (dense) Zariski open subset.
Show that any normal affine curve is smooth.
Show that any Zariski closed subset of ${\mathbb{P}}^n$ is compact in the Hausdorff topology.
Show that any projective variety is irreducible.
Show that the singular locus $X^{\mathrm{sing}}$ of a projective variety is a proper Zariski closed subset, and that if $X$ is normal, every irreducible component of $X^{\mathrm{sing}}$ has codimension $d'\geq 2$.
Show that a normal projective curve is smooth.
Show that if $X$ is projective and $Y$ is affine over $k = {\mathbb{C}}$, then the projection $\pi_2: X\times Y\to Y$ is proper and closed in the Zariski topology.
Let $f:X\to Y$ with $X, Y$ projective varieties over $k= {\mathbb{C}}$.
- Show that $f$ is proper and closed.
- Show that $f(X)$ is a projective subvariety.
- Show that if $f$ is dominant or birational, then $f$ is surjective and any regular function on $X$ is constant.
- Show that if $U \subseteq X$, then $\operatorname{ff}{\mathcal{O}}_U = {\mathbb{C}}(X)$.
- $f$ is birational iff $f^*: {\mathcal{O}}_Y\to {\mathcal{O}}_X$ is an isomorphism.
- $X\times Y$ is projective, using the Segre embedding.
For $X = V(f) \subseteq {\mathbb{P}}^3$ with $f$ irreducible homogeneous, show that $X$ is singular at $p$ iff $\operatorname{grad}f(p) = \mathbf{0}$.
- If $p$ is smooth, show that the tangent line has the equation $\operatorname{grad}f (p) \cdot {\left[ {x,y,z} \right]} = 0$.
Do isomorphic varieties have isomorphic affine cones?
Show that $\mathop{\mathrm{Aut}}{\mathbb{P}}^n{}_{/ {k}} = \operatorname{PGL}_{n+1}{}_{/ {k}} = \operatorname{GL}_n{}_{/ {k}} /{\mathbb{G}}_m$.
Show that a proper morphism between smooth projective curves is an isomorphism.
Let $\tilde X$ be the projective closure of $X$ and show $\tilde X = X\cup{{\partial}}X$, where ${{\partial}}X = \left\{{x_0 = 0}\right\} \cap X$.
What is $H^* {\mathbb{P}}^n_{/ {k}} $?
Show that any smooth cubic in ${\mathbb{P}}^2$ is an elliptic curve.
Is every smooth projective curve of genus 0 defined over the field of complex numbers isomorphic to a conic in the projective plane?
What is the maximum number of ramification points that a mapping of finite degree from one smooth projective curve over C of genus 1 to another (smooth projective curve of genus 1) can have?
Find an everywhere regular differential $n{\hbox{-}}$form on ${\mathbb{A}}^n$.
Prove that the canonical bundle of ${\mathbb{P}}^n$ is ${\mathcal{O}}(n-1)$
Show that if $X$ is a connected complete variety, then ${\mathcal{O}}_X(X) = k$, i.e. every global regular function is constant.

Examples

Show that $V(x^2+y^2+z^2,xyz)$ is a union of 6 lines.
Show that $\operatorname{GL}_n({\mathbb{C}})$ is an affine variety.
Show that ${\mathbb{A}}^2_{/ {{\mathbb{C}}}} \setminus\left\{{0}\right\}$ is not an affine variety.
Show that the Zariski topology on ${\mathbb{A}}^2_{/ {k}} $ is not the product topology on ${\mathbb{A}}^1_{/ {k}} \times {\mathbb{A}}^1_{/ {k}} $.
Show that the affine cubic $X = V(x(xy-1))$ is reducible and has two irreducible components.
Show that ${\mathbb{A}}^1$ is not isomorphic to $X = V(xy-1)$
- Show that $X$ has two connected components in the Hausdorff topology but is irreducible in the Zariski topology.
Consider the morphism \begin{align*} f: {\mathbb{A}}^2 &\to {\mathbb{A}}^2 \\ (x, y) &\mapsto (x, xy) .\end{align*} Is this finite? Dominant? Open? Closed? What are the fibers?
Cusps: show that the cuspidal cubic $X = V(x^3-y^2)$ has a unique singular point.
- Show that the normalization of $X$ is ${\mathbb{A}}^1$, using the birational map $t\mapsto (t^2,t^3)$.
- Show that the defining polynomial is irreducible in ${\mathbb{C}} { \left[ {x, y} \right] } $. What does this mean in terms of branching?
Double points: show that the nodal cubic $X = V(x^2 - y^2(y-1))$ has a unique singular point.
- Show that $X$ locally has two smooth branches at zero meeting transversally.
- Consider the projectivization $\tilde X = V(x^2z - y^2(y-z))$. Show that the normalization morphism $\nu: {\mathbb{P}}^1\to \tilde X$ is birational, and that $\nu^{-1}(0:0:1)$ is two points. Compare this to Zariski’s main theorem.
Write $\mu_n = \left\{{(x, y)\mapsto (\zeta_n x, \zeta_n y)}\right\}$ and define $X = \operatorname{mSpec}{\mathcal{O}}_{{\mathbb{A}}^2}^{\mu_n}$ be the subalgebra of $\mu_d$ invariants, equivalently $X = {\mathbb{A}}^2/\mu_n$. Show that $X$ is a normal affine variety with a unique singular point.
Show that the image of the Segre embedding ${\mathbb{P}}^1\times{\mathbb{P}}^1 \to {\mathbb{P}}^3$ is the smooth quadric $V(x_0x_3 - x_1 x_2)$.
Define the Weierstrass cubic as $X\coloneqq V(y^{2} z-(x^{3}+g_{2} x z^{2}+g_{3} z^{3}) )$ for $g_2, g_3 \in {\mathbb{C}}$. Show that $X$ is nonsingular iff $p(x) \coloneqq x^3 + g_2 x + g_3$ has no multiple roots.
Let $X = V(xy-z^2)$ and compute the class group of $X$.
Let $X = V(y^2-x(x^2-1))$ and compute the class group of $X$.
Compute ${\operatorname{Pic}}{\mathbb{P}}^n_{/ {k}} $.