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date: 2022-12-29 21:10
modification date: Thursday 29th December 2022 21:10:09
title: "Unused Oral Exam Questions"
aliases: [Unused Oral Exam Questions]
---

Last modified date: NaN

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	- #todo/untagged
- Refs:
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- Links:
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# Unused Oral Exam Questions


# Unusued

- [ ] What is the **normalization** of an affine variety?
- [ ] What is the **Segre embedding**?
- [ ] What is the **Veronese embedding**?
- [ ] What is the **affine cone** on a projective variety?
- [ ] What is the **projective closure** of an affine variety
- [ ] Given a projective variety $X$, what is its **affine cone**?
- [ ] What is the **Hilbert function** of a variety? The **Hilbert polynomial**?
- [ ] What is the **rational function field** $\kappa_X$ of an affine variety?
- [ ] What does it mean for elements of $\kappa_X$ to be **algebraically independent**?
- [ ] What is the hyperplane at $\infty$ of ${\mathbb{P}}^n$ in coordinates?
- [ ] What is the **degree of a $k{\hbox{-}}$dimensional subvariety** of ${\mathbb{P}}^n$?
- [ ] What is the **Néron-Severi group**?
- [ ] What is the **canonical bundle**? The **anticanonical**? **Pluricanonical** bundles?
- [ ] What is the **tautological bundle**?
- [ ] What does it mean for a bundle to be **positive**?
- [ ] What is a **cuspidal singularity**?
- [ ] What is a **nodal singularity**?
- [ ] What is a **normal crossing singularity**?
	- [ ] For a divisor, each irreducible component is smooth, and whenever $r$ irreducible $Y_i$ components intersect at $p$, the local equations $f_j$ of the $Y_i$ form a regular system of parameters at $p$, i.e. $f_1,\cdots, f_r$ are linearly independent $\mod \mfm_p^2$.
- [ ] What is an **embedding** of projective varieties?
- [ ] What is the **exceptional curve/divisor**?
- [ ] What is a **Chow variety**?
- [ ] What is the **Grassmannian**?
- [ ] What is a **tangent cone**?
- [ ] What is a **dual variety**?
- [ ] What is a **hypersurface**?
- [ ] What is a **very general hypersurface**?
- [ ] What does it mean to be **unirational**?
- [ ] What does it mean to be **ruled**?
- [ ] What does it mean to be **uniruled**?
- [ ] What is a **scroll**?
- [ ] What is the **gonality** of a curve?
- [ ] What is a **linear series**?
- [ ] What is **specialization**?
- [ ] What is a **regular function**?
- [ ] What is a **finite** morphism?
- [ ] What is a **quasi-finite** morphism?
- [ ] What is a **closed** morphism?
- [ ] What is a **birational morphism**?
- [ ] What is a **rational function** on a variety?
- [ ] What is the **degree** of a morphism?
- [ ] What is a **rational map** of affine varieties?
- [ ] What is the **indeterminacy set** of a rational map?
- [ ] What is the exceptional inverse image sheaf?
- [ ] What is the Leray map?
- [ ] What is [Kodaira vanishing](Kodaira%20vanishing.md)?
- [ ] What is [Serre's projection formula](Serre’s%20projection%20formula)?
- [ ] What is [Serre's finiteness theorem](Serre’s%20finiteness%20theorem)?
- [ ] What is [Chevalley's finiteness theorem](Chevalley’s%20finiteness%20theorem)?
- [ ] What is [Hensel's lemma](Hensel’s%20lemma)?
- [ ] What are the [weak and hard Lefschetz theorems](weak%20and%20hard%20Lefschetz%20theorems.md)?
- [ ] What is the [Jacobi criterion](Jacobi%20criterion)?
- [ ] What is an example of an [unramified morphism](unramified%20morphism.md)?
- [ ] When is the [canonical sheaf](canonical%20sheaf) dualizing?
- [ ] What is the cohomology of $O_X$ for $X = {\mathbb{P}}^n$
- [ ] What is an example of a scheme that is not a variety?
- [ ] Show that the affine line with two origins is not separated.
- [ ] Show that $X$ is a projective variety over $\operatorname{Spec}k$ iff $X$ is a closed subscheme of $\PP^n\slice k$.
- [ ] Show that ${\mathsf{R}{\hbox{-}}\mathsf{Mod}} { \, \xrightarrow{\sim}\, }{\mathsf{QCoh}}(\operatorname{Spec}R)$, and that ${\mathsf{Coh}}(\operatorname{Spec}R)$ corresponds to finitely-generated modules.
- [ ] Show that a locally free sheaf is always quasicoherent, and is coherent if it has finite rank.
- [ ] Show that if ${\mathcal{L}}$ is a line bundle, then ${\mathcal{L}}\otimes_{{\mathcal{O}}_X} {\mathcal{L}} {}^{ \vee } { \, \xrightarrow{\sim}\, }{\mathcal{O}}_X$.
- [ ] Show that $\left\{{f_i}\right\}$ generate the unit ideal in $R$ iff $D(f_i) \rightrightarrows\operatorname{Spec}R$.
- [ ] Show that $M$ is a faithfully flat module iff ${-}\otimes_R M$ is a [faithful functor](faithful%20functor.md).
- [ ] Show that a finite morphism between smooth varieties is flat.
- [ ] Show that a regular function on an affine scheme is not necessarily determined by its values.
- [ ] Show that the hyperplane sections of a projective variety $X$ form a base point free linear system of effective divisors on $X$.
- [ ] Show that if $X$ is normal, then a generic element of this system is a smooth and reduced divisor.
- [ ] Show that ${ \operatorname{Cl}} ({\mathbb{A}}^n_{/ {k}} ) = 0$
- [ ] Show that ${ \operatorname{Cl}} ({\mathbb{P}}^n_{/ {k}} ) \cong {\mathbb{Z}}$, generated by the class of $H = V(x_i)$.
- [ ] Show that geometric genus, plurigenus, and irregularity are birational invariants.

- [ ] What is a **soft** sheaf?
- [ ]  What is the **Godemont resolution**?
- [ ] What is the **Chow ring**?
- [ ] What is the **cycle class map**?
- [ ] What is the [canonical class](canonical%20class)?
- [ ] What is the [cotangent complex](Unsorted/cotangent%20complex.md)?
- [ ] What is a [constructible sheaf](constructible%20sheaf.md)?
- [ ] Why are higher direct images with proper support of l-adic sheaves again l-adic?
- [ ] Why is the [moduli space of curves](moduli%20space%20of%20curves) ${\mathcal{M}}_g$ smooth and proper?
- [ ] Show that the differential in the de Rham complex is not ${\mathcal{O}}_X{\hbox{-}}$linear.
- [ ] Compute the Cech cohomology of $\PP^n\slice k$.
- [ ] Compute $H^*(X, {\mathcal{O}}_X)$ for $X=\PP^n\slice k$.
- [ ] For $X\to B$ smooth, why does $\operatorname{Ext} ^0(\Omega_{X/B}, {\mathcal{O}}_X) \cong H^0({\mathbf{T}}_{X/B} )$ hold, and why does it measure infinitesimal automorpisms of $X$?
- [ ] What does $\operatorname{Ext} ^1(\Omega_{X/B}, {\mathcal{O}}_X)$ measure 1st order deformations of $X\to B$?
- [ ] Why doesn't dualizing the [cotangent complex](Unsorted/cotangent%20complex.md) yield the tangent complex? What is the correct dualization to take?
- [ ] Show that singular cohomology is isomorphic to Cech cohomology for Noetherian separated schemes.
- [ ] Show that the dualizing sheaf is isomorphic to the canonical sheaf for nonsingular projective varieties.
- [ ] What is the **semicontinuity theorem**?
- [ ] How is the sheaf of differentials related to the singularities/smoothness of a scheme?
- [ ]   What is the [Lefschetz hyperplane theorem](Lefschetz%20hyperplane%20theorem)
- [ ]   What is [Hodge duality](Hodge%20duality)?
- [ ] What does "cohomology commutes with flat base extension" mean?
- [ ] What is [Castelnuovo's theorem](Castelnuovo’s%20theorem)
- [ ] What is the [Brill-Noether theorem](Brill-Noether%20theorem.md)?
- [ ] Why is every genus 2 curve [hyperellptic](hyperellptic)? Why is this not even generically true for $g\geq 3$?
- [ ] How can you view a curve as a [ramified cover](ramified%20cover) of $\PP^1$?
- [ ] What is the [cohomological criterion for ampleness](cohomological%20criterion%20for%20ampleness.md)?
- [ ] Why is [Zariski's theorem](Zariski's%20theorem.md) true?
- [ ] What is the [Hodge index theorem](Hodge%20index%20theorem)?
- [ ] What is [Mori's theorem](Mori's%20theorem)?
- [ ] What is the [minimal model program](minimal%20model%20program.md) for surfaces?

- [ ]   Show that if $C$ is a smooth proper curve of genus $g(C)\geq 2$, then ${\sharp}\mathop{\mathrm{Aut}}(G) < \infty$.
- [ ]   Show that given $\pi: X \rightarrow Y$, $Y$ is integrally closed in $K(X)$ over $K(Y)$ if and only if ${\mathcal{O}}_{Y} \rightarrow \pi_{*} {\mathcal{O}}_{X}$ is an isomorphism (i.e., $\pi$ is ${\mathcal{O}}$-connected).
- [ ]   Use [Tsen's theorem](Tsen's%20theorem.md) to show that given a flat family $X\to Y$ with $Y$ a curve where the fibers smooth curves of genus 0, this determines a Zariski ${\mathbb{P}}^1{\hbox{-}}$bundle iff there exists a relative degree 1 line bundle on $X$ over $Y$.

- [ ] Show that the following presheaves need not be sheaves:
- [ ] The tensor presheaf $U\mapsto {\mathcal{F}}(U)\otimes_{{\mathcal{O}}_X(U)} {\mathcal{G}}(U)$.
- [ ] The image presheaf $U \mapsto \operatorname{im}({\mathcal{F}}(U) \to {\mathcal{G}}(U))$
- [ ] Show that the cokernel presheaf $U\mapsto \operatorname{coker}({\mathcal{F}}(U) \to {\mathcal{G}}(U))$.
- [ ] Let $X = {\mathbb{P}}^1$.
- [ ] Show that the skyscraper sequence is exact: $$0 \to {\mathcal{O}}_{{\mathbb{P}}^1}(-1) \to {\mathcal{O}}_{{\mathbb{P}}^1} \to K_{{\left[ {1: 0} \right]}} \to 0$$
- [ ] Find $\operatorname{coker}({\mathcal{O}}_X(-2) \to {\mathcal{O}}_X)$
- [ ] Show that ${\mathcal{O}}_X(d_1) \otimes{\mathcal{O}}_X(d_2) { \, \xrightarrow{\sim}\, }{\mathcal{O}}_X(d_1+d_2)$.
- [ ] Find $\operatorname{coker}({\mathcal{O}}_X \to {\mathcal{O}}_X(1){ {}^{ \scriptscriptstyle\oplus^{2} } })$.
- [ ] Show that ${\mathcal{O}}_X(d) {}^{ \vee } { \, \xrightarrow{\sim}\, }{\mathcal{O}}_X(-d)$.
- [ ] Produce a sequence that is exact on sheaves but not exact on the corresponding presheaves.
- [ ] Show that a pushforward of a locally constant sheaf need not be a locally constant sheaf, and conclude that the stalks do not entirely determine a sheaf.
	- [ ] ![](attachments/Pasted%20image%2020220315153140.png)
	- [ ] ![](attachments/Pasted%20image%2020220315153154.png)
- [ ] Show that pushforward along $f$ is left exact, and is fully exact if $f: U\hookrightarrow X$ is injective with closed image.
- [ ] Show that if $i: \left\{{x}\right\}\hookrightarrow X$ then $i^{-1}{\mathcal{F}}_x \cong {\mathcal{F}}_x$
- [ ] Show that if $X$ is a Noetherian separated scheme with the Zariski topology, ${\mathcal{F}}\in{\mathsf{QCoh}}(X)$, and ${\mathcal{U}}$ is any open cover, then the Leray map is an isomorphism ${ {{\check{H}}}^{\scriptscriptstyle \bullet}} ({\mathcal{U}};{\mathcal{F}}) { \, \xrightarrow{\sim}\, } { {H}^{\scriptscriptstyle \bullet}} (X;{\mathcal{F}}) \coloneqq { {{\mathbb{R}}}^{\scriptscriptstyle \bullet}} \Gamma(X; {\mathcal{F}})$