# Varieties: Problems

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