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Problem 0: Introduction
My plan is to post some daily AG problems from various sources, although they will mainly be from Hartshorne, Beauville’s Complex Algebraic Surfaces, Fulton’s Introduction to Toric Varieties, and Kollár’s…
NB 1: Hartshorne marks the problems that may be more difficult with a *. I’ve decided to mark problems that I found most difficult with a ^. Almost always, * implies ^, but the converse certainly need not hold.
NB 2: T5 favorite (2,9,14,29,30) | HM (7,24,27) | T5 least favorite (HA II.3.3,II.3.4)
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Problem 48: Hartshorne Exercise II.6.9
*Hartshorne Exercise II.6.9: Singular Curves Here we give another method of calculating the Picard group of a singular curve. Let $X$ be a projective curve over $k$, let $X$ be…
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Problem 47: Lazarsfeld’s PAG Example 1.2.30
Problem 47: Lazarsfeld Example 1.2.30: Divisors with ample normal bundle. Again, Lazarsfeld mostly leads the way with these examples. It has been helpful for me to work through the details…
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Problem 46: Lazarsfeld Positivity in AG, Example 1.2.22.
Lazarsfeld’s Positivity in Algebraic Geometry: Example 1.2.22 I’m currently reading Lazarsfeld’s book, Positivity in Algebraic Geometry. It is really a joy to read, and I plan to do related exercises…
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“Problem” 45. Notes on the pullback and pushforward.
Currently, there are several problems I am working on; I seem to be stuck on all of them, for the time being, but I feel like I am close to…
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Problem 44: Hartshorne II.5.12
Hartshorne Exercise II.5.12. (a) Let $X$ be a scheme over a scheme $Y$, and let $\mathcal L,\mathcal M$ be two very ample invertible sheaves on $X$. Show that $\mathcal L\otimes\mathcal…
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Problem 42: Hartshorne II.5.13
Hartshorne Exercise II.5.13. Let $S$ be a graded ring, generated by $S_1$ as an $S_0$ -algebra. For any integer $d>0$, let $S^{(d)}$ be the graded ring $\bigoplus_{n\ge 0} S_{n}^{(d)}$ where…
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Problem 41: Hartshorne II.6.8
Hartshorne Exercise II.6.8. Let $\varphi: \PP{k}{n}\ra\PP{k}{m}$ be a morphism. Then: (a) Let $f:X\ra Y$ be a morphism of schemes. Show that $\mathcal L\mapsto f^*\mathcal L$ induces a homomorphism of Picard…
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Problem 40: Harthsorne III.5.1
Hartshorne Exercise III.5.1. Let $X$ be a projective scheme over a field $k$, and let $\mathcal F$ be a coherent sheaf on $X$. We define the Euler Characteristic of $\mathcal…
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Problem 39: Harshorne II.7.8
Hartshorne Exercise II.7.8. Let $X$ be a noetherian scheme, let $\mathcal E$ be a coherent locally free sheaf on $X$, and let $\pi: \Bbb P(\mathcal E)\ra X$ be the corresponding…
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Problem 38: Hartshorne II.7.7.
Hartshorne Exercise II.7.5. Some Rational Surfaces. Let $X=\PP{k}{2}$, and let $|D|$ be the complete linear system of all divisors of degree 2 on $X$ (conics). $D$ corresponds to the invertible…
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Problem 37: Hartshorne II.7.3.
Hartshorne Exercise II.7.3. Let $\varphi: \PP{k}{n}\ra\PP{k}{m}$ be a morphism. Then: (a) either $\varphi(\PP{k}{n})={\op{pt}}$ or $m\ge n$ and $\dim\varphi (\PP{k}{n})=n$;(b) in the second case, $\varphi$ can be obtained as the composition…
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Problem 36: Hartshorne II.7.5.
Hartshorne Exercise II.7.5. Establish the following properties of ample and very ample invertible sheaves on a noetherian scheme $X$. $\mathcal L,\mathcal M$ will denote invertible sheaves, and for (d), (e)…
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Problem 35: Hartshorne II.7.4.
Hartshorne Exercise II.7.4. (a) Use (7.6) to show that if $X$ is a scheme of finite type over a noetherian ring $A$, and if $X$ admits an ample invertible sheaf,…
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Problem 34: Hartshorne II.7.2.
After a six-week moratorium, we are back. I plan to keep posting problems and solutions daily, although this plan may be overly ambitious this quarter due to grad school applications.…
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Problem 32：Resolution of DuVal Singularities Part 1.
Exercise: Resolve the 5 forms of DuVal Singularities (Part 1/5: $A_n$ singularities). The Minimal Model Program has specific conventions in the ways in which it classifies singularities on varieties. In…
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Problem 31: Hartshorne V.1.3.
Hartshorne Exercise V.1.3. Recall that the arithmetic genus of a projective scheme D of dimension 1 is defined as $p_a=1-\chi(\OO_D)$ (III, Ex. 5.3). (a) If $D$ is an effective divisor…
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Problem 30: Hartshorne II.6.5
Hartshorne Exercise II.6.5. Quadric Hypersurfaces. Let $\op{char} k \ne 2$, and let $X$ be the affine quadric hypersurface $\spec{ k[x_0,…,x_n]/(x_0^2+…+x_r^2)}$ cf. (I, Ex. 5.12).(a) Show that $X$ is normal if…
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Problem 29: Hartshorne II.6.3*
Hartshorne Exercise II.6.3. Cones. In this exercise we compare the class group of a projective variety $V$ to the class group of its cone (I,Ex.2.10). So let V be a…
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Problem 27: Hartshorne II.3.7
Hartshorne Exercise II.3.7. A morphism $f:X \ra Y$, with Y irreducible, is generically finite if $f^{-1}(\eta)$ is a finite set, where $\eta$ is the generic point of Y. A morphism…
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Problem 26: Hartshorne II.6.2*
Hartshorne Exercise II.6.2. 6.2. Varieties in Projective Space. Let $k$ be an algebraically closed field, and let $X$ be a closed subvariety of $\PP{k}{n}$ which is nonsingular in codimension one…
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Problem 25: Hartshorne II.6.4.
Hartshorne Exercise II.6.4. Let $k$ be a field of characteristic $\ne 2$. Let $f\in k[x_1,…,x_n]$ be a squarefree nonconstant polynomial, i.e., in the unique factorization of $f$ into irreducible polynomials,…
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Problem 24: Hartshorne II.6.1
Hartshorne Exercise II.6.1. Let $X$ be a scheme satisfying $(*)$. Then $X\times\PP{k}{n}$ also satisfies $(*)$,and $\op{Cl}(X\times\PP{k}{n})\cong \op{Cl}X\times \Z$. Let $\pi_1,\pi_2$ be the first and second projections of $X\times\PP{k}{n}$. Clearly $X\times\PP{k}{n}$…
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Problem 23: Hartshorne II.7.1
Hartshorne Exercise II.7.1. Let $(X,\OO_X)$ be a locally ringed space, and let $f:\mathcal{L}\ra\mathcal{M}$ be a surjective map of invertible sheaves on $X$. Show that $f$ is an isomorphism. [Hint: Reduce…
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Problem 21: Beauville II.20
Beauville Exercise II.20. Let C be curve on a smooth, projective surface S. Show that there is a morphism of smooth surfaces $\tilde{S}\rightarrow S$ such that the strict transform $\tilde{C}\subset\tilde{S}…
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Problem 20: Hartshorne IV.1.6
Hartshorne Exercise IV.1.6. Let $X$ be a curve of genus $g$. Show that there is a finite morphism $f:X\ra\PP{k}{1}$ of degree $\le g+1$. (Recall that the degree of a finite…
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Problem 19: Hartshorne IV.1.5
Hartshorne Exercise IV.1.5. For an effective divisor $D$ on a curve $X$ of genus $g$, show that $\dim |D|\le\deg D$. Furthermore, equality holds if and only if $D=0$ or $g=0$.…
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Problem 18: Hartshorne IV.1.3
Hartshorne Exercise IV.1.3. Let $X$ be an integral, separated, regular, one-dimensional scheme of finite type over $k$, which is not proper over $k$. Then $X$ is affine. [Hint: Embed $X$…
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Problem 17: Hartshorne IV.1.2
Hartshorne Exercise IV.1.2. Again let $X$ be a curve, and let $P_1,…,P_r\in X$ be points. Then there is a rational function $f\in \Bbb K(X)$ having poles (of some order) at…
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Problem 16: Hartshorne IV.1.1
Hartshorne Exercise V.1.1. Let $X$ be a curve, and let $P\in X$ be a point. Then there exists a nonconstant rational function $f\in \Bbb{K}(X)$, which is regular everywhere except at…
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Problem 15: Hartshorne V.1.4
Hartshorne Exercise V.1.4. (a) If a surface $X$ of degree $d$ in $\PP{k}{3}$ contains a straight line $C=\PP{k}{1}$, show that $C^2=2-d$. (b) Assume $\op{char}k=0$, and show for every $d\ge 1$,…
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Problem 14: Hartshorne V.1.11
Hartshorne Exercise V.1.11.In this problem, we assume that $X$ is a surface for which $\op{Num}X$ is finitely generated (i.e., any surface, if you accept the Neron~Severi theorem (Ex. 1.7) ).…
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Problem 13: Hartshorne V.1.9
Hartshorne Exercise V.1.9. (a) If $H$ is an ample divisor on a surface $X$, and $D$ is any divisor, show that $$(D^2)(H^2)\le(D.H)^2.$$ (b) Now let $X$ be a product of…
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Problem 12: Hartshorne V.3.3
Hartshorne Exercise V.3.3. Let $\pi:\tilde X\ra X$ be a monoidal transformation, and let $D$ be a very ample divisor on $X$. Show that $2\pi^*D-E$ is ample on $\tilde X$. [Hint:…
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Problem 11: Hartshorne V.3.2
Hartshorne Exercise V.3.2. Let $C$ and $D$ be curves on a surface $X$, meeting at a point $P$. Let $\pi:\tilde X\ra X$ be the monoidal transformation with center $P$. Show…
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Problem 10: Hartshorne V.1.6
Hartshorne Exercise V.1.5 (a) If $C$ is a curve of genus $g$, show that the diagonal $\Delta\subseteq C\times C$ has self intersection $\Delta^2=2-2g.$ (b) Let $l=C\times\text{pt}$ and $m=\text{pt}\times C.$ If…
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Problem 9: Hartshorne V.1.5
Hartshorne Exercise V.1.5 (a) If $X$ is a surface of degree $d$ in $\PP{k}{3}$, then $K^2=d(d-4)^2$. (b) If $X$ is a product of two nonsingular curves $C,C’$ of genus $g,g’$…
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Problem 8: Hartshorne V.1.2
Hartshorne Exercise V.1.2 Let $H$ be a very ample divisor on a surface $X$, corresponding to a projective embedding $X\subseteq\PP{k}{N}$. If we write the Hilbert polynomial of $X$ (III, Ex.…
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Problem 7: Hartshorne III.5.2
Hartshorne Exercise III.5.2. (a) Let $X$ be a projective scheme over a field $k$, let $\OO_X(1)$ be a very ample invertible sheaf on $X$ over $k$, and let $\mathcal{F}$ be…
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Problem 6: Hartshorne IV.3.3
Hartshorne Exercise IV.3.3. Let $X$ be a plane curve of degree 4. (a) Show that the effective canonical divisors on $X$ are exactly the divisors $X.L$, where $L$ is a…
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Problem 5: Hartshorne IV.3.1
Hartshorne Exercise IV.3.7. If $X$ is a curve of genus 2, show that a divisor $D$ is very ample $\iff$ $\deg D\ge 5$. Let $D$ be a divisor on a…
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Problem 4: Hartshorne II.5.8
Hartshorne Exercise II.5.8. Again let X be a noetherian scheme, and $\mathcal{F}$ a coherent sheaf on X. We will consider the function, $$\varphi(x)=\op{dim}_{k(x)}\mathcal{F}_x\otimes_{\OO_x}k(x)$$, where $k(x)=\OO_x/\mathfrak{m}_x$ is the residue field at…
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Problem 3: Hartshorne II.5.7
Hartshorne Problem II.5.7. Let X be a noetherian scheme, and let $\mathcal{F}$ be a coherent sheaf. (a) If the stalk $\mathcal{F}_x$ is a free $\OO_{X,x}$-module for some point $x\in X$,…
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Problem 2: Hartshorne Exercise IV.3.3 and some generalizations of Problem 1
We first make some interesting generalizations to the previous post, problem 1. Recall that we showed that the blowup of $\PP{k}{2}$ at $n$ different points remains a (rational) Fano surface…
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Problem 1: Show that the blowup of $\Bbb P_k^2$ at up to 8 points can be Fano.
Define inductively a sequence of blowups $\varepsilon_n:S_n\ra S_{n-1}$, where $\varepsilon_n:S_n\ra S_{n-1}$ is the blowup of $S_{n-1}$ at a point $p_{n-1}$ and $S_0:=\Bbb P_k^2$. Let $E_n$ be the exceptional divisor of…
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Problem 0: Introduction
My plan is to post some daily AG problems from various sources, although they will mainly be from Hartshorne, Beauville’s Complex Algebraic Surfaces, Fulton’s Introduction to Toric Varieties, and Kollár’s…
