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 ${\operatorname{Pic}}{\mathbb{P}}^n_{/ {k}}$.