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?
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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?
Problems
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Show that a projective variety \(X\) is irreducible iff \(I(V)\) is a (homogeneous) prime ideal
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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.
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Show that \({\mathcal{O}}_{{\mathbb{A}}^n}\) is Noetherian.
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Show that any Zariski open is dense.
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Show that the Zariski topology on \(X\) is never separated unless \(X\) is a point.
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Show that if \(X, Y\) are affine varieties, then \(X\times Y\) is an affine variety with \({\mathcal{O}}_{X\times Y} \cong {\mathcal{O}}_X \otimes_k {\mathcal{O}}_Y\).
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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.
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Show that any morphism \({\mathbb{A}}^1\to{\mathbb{A}}^1\setminus\left\{{0}\right\}\) is constant.
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Show that if \(f:X\to Y\) is quasi-finite, then \(\dim X \leq \dim Y\).
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Show that if \(f:X\to Y\) is finite and \(\dim X = \dim Y\), then \(f\) is closed and surjective.
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Can a non-surjective finite morphism exist?
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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 \(Y\).
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Show that if \(f\in {\mathcal{O}}_X\) be nonconstant, then any irreducible component of the fiber \(f^{-1}(0)\) has dimension \(d = \dim X - 1\).
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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 \(\dim {\mathbf{T}}_p X > d\).
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Show that a point \(p\in X\) is smooth iff \(X \hookrightarrow{\mathbb{A}}^n\) is locally a smooth submanifold.
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Show that \({\mathbf{T}}_p X \cong ({\mathfrak{m}}_p/{\mathfrak{m}}_p^2) {}^{ \vee }\).
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Show that the singular points of \(X\) form a proper Zariski closed subset, so smooth points form a (dense) Zariski open subset.
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Show that any normal affine curve is smooth.
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Show that any Zariski closed subset of \({\mathbb{P}}^n\) is compact in the Hausdorff topology.
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Show that any projective variety is irreducible.
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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 \(d'\geq 2\).
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Show that a normal projective curve is smooth.
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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.
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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.
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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 \(\operatorname{grad}f (p) \cdot {\left[ {x,y,z} \right]} = 0\).
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Do isomorphic varieties have isomorphic affine cones?
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Show that \(\mathop{\mathrm{Aut}}{\mathbb{P}}^n{}_{/ {k}} = \operatorname{PGL}_{n+1}{}_{/ {k}} = \operatorname{GL}_n{}_{/ {k}} /{\mathbb{G}}_m\).
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Show that a proper morphism between smooth projective curves is an isomorphism.
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Let \(\tilde X\) be the projective closure of \(X\) and show \(\tilde X = X\cup{{\partial}}X\), where \({{\partial}}X = \left\{{x_0 = 0}\right\} \cap X\).
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What is $H^* {\mathbb{P}}^n_{/ {k}} $?
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Show that any smooth cubic in \({\mathbb{P}}^2\) is an elliptic curve.
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Is every smooth projective curve of genus 0 defined over the field of complex numbers isomorphic to a conic in the projective plane?
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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?
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Find an everywhere regular differential \(n{\hbox{-}}\)form on \({\mathbb{A}}^n\).
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Prove that the canonical bundle of \({\mathbb{P}}^n\) is \({\mathcal{O}}(n-1)\)
Examples
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Show that \(V(x^2+y^2+z^2,xyz)\) is a union of 6 lines.
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Show that \(\operatorname{GL}_n({\mathbb{C}})\) is an affine variety.
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Show that \({\mathbb{A}}^2_{/ {{\mathbb{C}}}} \setminus\left\{{0}\right\}\) is not an affine variety.
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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}} $.
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Show that the affine cubic \(X = V(x(xy-1))\) is reducible and has two irreducible components.
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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.
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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?
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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?
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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.
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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.
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Show that the image of the Segre embedding \({\mathbb{P}}^1\times{\mathbb{P}}^1 \to {\mathbb{P}}^3\) is the smooth quadric \(V(x_0x_3 - x_1 x_2)\).
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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.
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Let \(X = V(xy-z^2)\) and compute the class group of \(X\).
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Let \(X = V(y^2-x(x^2-1))\) and compute the class group of \(X\).
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Compute ${\operatorname{Pic}}{\mathbb{P}}^n_{/ {k}} $.