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Since $g$ is a separated morphism, the diagonal morphism ${\Delta }_g:Y\to Y{\times }_{Z}Y$ is a closed immersion. The squareis cartesian. Indeed, on test schemes its pullback condition is $(f(x),y)=(t,t)$, forcing $y=t=f(x)$, and leaving exactly the choice of $x$. The graph morphism of schemes ${\Gamma }_f$ is therefore a base change of a morphism of schemes of ${\Delta }_g$. Part (c) shows that ${\Gamma }_f$ is a closed immersion.

Now assume additionally that $h=g\circ f:X\to Z$ is a closed immersion. The projection $p_Y:X{\times }_{Z}Y\to Y$ is its base change of a morphism of schemes along $g$, so is a closed immersion by part (c). Sincepart (b) shows that $f$ is a closed immersion. This covers the further assumption in the final unheaded paragraph.

A line bundle $\mathcal{L}$ is a globally generated line bundle when the evaluation mapis surjective. Equivalently, its global sections span each fibre of the line bundle; locally at every point, some section is a generator.

Choose a finite generating family $s_0,\dots ,s_n$. On the open set where $s_i$ generates $\mathcal{L}$, the ratios $s_j/s_i$ are regular functions, giving a morphism of schemes into the standard chart $D_{+}(x_i)$ of projective space. The ratios obey the usual transition rules on overlaps, so these chart maps glue toThe tuple is computed using any local trivialization of $\mathcal{L}$; changing that trivialization multiplies all entries by the same invertible function. The construction has ${\phi }^{\ast }\mathcal{O}(1)\simeq \mathcal{L}$ and pulls back the coordinate sections to the chosen $s_i$.

There is a finiteness qualification for an arbitrary $X$: global generation alone need not provide such a finite family. It does if $X$ is quasi-compact, since the open sets on which individual sections generate have a finite subcover. Without that hypothesis, take $X={\coprod }_{m\ge 1}{\mathbb{P}}_k^m$ and let $\mathcal{L}$ restrict to $\mathcal{O}(1)$ on each component. This line bundle is globally generated, but any $N$ global sections have a common zero on a component ${\mathbb{P}}^m$ with $m\ge N$. Thus this is a globally generated line bundle without finite generators. This is finite global generation on a quasi-compact scheme. The finite-family construction is automatic in the projective case asked next.

For projective nonsingular $X$, put $V=⟨s_0,\dots ,s_n⟩$. The length-two criterion for a very ample linear system says that $\phi$ is a closed immersion precisely when, after extending to an algebraic closure, $V$ separates distinct points and tangent directions. Equivalently, the evaluationis surjective for every length-two geometric closed subscheme $E\subseteq X$. Two distinct points give point separation; a nonreduced length-two subscheme supported at one point gives separation of a direction in the Zariski tangent space. For the complete space $V=H^0(X,\mathcal{L})$, this says exactly that $\mathcal{L}$ is a very ample line bundle. For a chosen smaller $V$, it is the chosen linear system of divisors which must be a very ample linear system; mere global generation is insufficient.

Here $\mathrm{deg}{K}_C=2$. For $D$, $D-P$ and $D-P-Q$, including $Q=P$, the degrees are $5,4,3$, all greater than two. Thus the correction term in the Riemann-Roch theorem vanishes andThe first drop supplies a section nonzero at every chosen point, proving base-point-freeness of a linear system. For distinct $P,Q$, the second drop supplies a section vanishing at one but not the other, proving point separation. For $Q=P$, it supplies a section vanishing to order exactly one, proving tangent separation. Hence the complete linear system of divisors is very ample and $\boxed{C\hookrightarrow {\mathbb{P}}^3}$ is the induced closed immersion.

For a smooth projective algebraic curve, a linear system of divisors comes from a finite-dimensional space $V\subseteq H^0(C,\mathcal{O}(D))$. A basis of sections $s_0,\dots ,s_n$ defines $f(P)=[s_0(P):\cdots :s_n(P)]$, using any local trivialization; a common scalar change does not alter the projective point. A base-point-free linear system makes this defined everywhere, and $f^{\ast }{\mathcal{O}}_{{\mathbb{P}}^n}(1)\cong \mathcal{O}(D)$. Pulling back a hyperplane gives the zero divisor of a linear combination of the sections, an element of the original system.

A system with base divisor $B$ can instead define a map after removing $B$, but its hyperplane pullbacks have class $D-B$. Thus base-point-freeness is needed for the stated original-system property. For a closed immersion, the system must also separate distinct points and tangent directions; equivalently it is a very ample linear system. On a smooth curve the latter tangent condition is separation of the length-two subscheme at each point.

For an affine open subscheme $U=\mathrm{Spec}R$, quasi-coherence gives $\mathcal{I}{|}_U=\tilde{J}$ for an ideal $J\subseteq R$. Define $Z_U=\mathrm{Spec}(R/J)$ with its usual closed immersion into $U$. On a principal open subscheme $D(a)$, its restriction is $\mathrm{Spec}(R_a/J_a)$, because localization commutes with taking this quotient. These local constructions therefore agree on overlaps and glue.

The resulting underlying closed set is $Z=\{x:{\mathcal{I}}_x\ne {\mathcal{O}}_{X,x}\}$, locally $V(J)$. With inclusion $i:Z\hookrightarrow X$, its structure sheaf isAt a point corresponding to $\mathfrak{p}\supseteq J$, its stalk is $R_{\mathfrak{p}}/J_{\mathfrak{p}}$, the local ring of $\mathrm{Spec}(R/J)$. Thus the glued ringed space is a scheme and $i$ is a closed immersion, with ideal sheaf of a closed subscheme exactly $\mathcal{I}$. This is the construction of a closed subscheme from a quasi-coherent ideal. If $\mathcal{I}={\mathcal{O}}_X$, the construction gives the empty scheme; if $\mathcal{I}=0$, it gives $X$.

The resolution principle for sheaf cohomology states that an exact sequencewith $H^q(X,{\mathcal{A}}^j)=0$ for every $q>0$ computes $H^i(X,\mathcal{F})$ as the cohomology of $\Gamma (X,{\mathcal{A}}^{\bullet })$. In particular one may use a flasque resolution.

An affine morphism $\varphi :X\to Y$ is one for which the inverse image of every affine open subset of $Y$ is affine. We use vanishing of quasi-coherent cohomology on an affine scheme: a quasi-coherent sheaf has zero higher sheaf cohomology on an affine variety. Take a flasque resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$. Each direct image sheaf ${\varphi }_{\ast }{\mathcal{I}}^j$ is flasque, because restriction on $V\subseteq U\subseteq Y$ is restriction on ${\varphi }^{-1}V\subseteq {\varphi }^{-1}U$. On an affine open $U\subseteq Y$, the augmented complex of sections isIts positive-degree cohomology vanishes by the affine theorem, and its degree-zero kernel is $\Gamma ({\varphi }^{-1}U,\mathcal{F})$. Checking exactness on this affine basis shows that ${\varphi }_{\ast }{\mathcal{I}}^{\bullet }$ is a flasque resolution of ${\varphi }_{\ast }\mathcal{F}$. The global-section complexes are identical, proving cohomology under an affine morphism:

For projective space, put $S=k[x_0,\dots ,x_n]$ with its usual grading and $U_i=\{x_i\ne 0\}$. Construct the twisting sheaf on projective space by setting its sections on $U_i$ to be the degree-$m$ part $(S_{x_i}{)}_m$, viewed as a module over $(S_{x_i}{)}_0$. It is free of rank one with generator $x_i^m$, even when $m$ is negative. On overlaps the generators differ by the unit $(x_i/x_j{)}^m$, so these modules glue to a line bundle $\mathcal{O}(m)$.

Assume first $n\ge 1$. These local modules all embed in the graded fraction field, and agreement on overlaps identifies global sections with ${\bigcap }_i(S_{x_i}{)}_m$. Since $S$ is a unique factorization domain and at least two distinct variables occur, a reduced fraction belonging to every $S_{x_i}$ has no nonconstant denominator. Thus the intersection is $S_m$. Counting monomials by stars and bars givesHere $h_n^i(m)={\dim }_k{H}^i({\mathbf{P}}^n,\mathcal{O}(m))$. The printed formula needs the convention that a counting binomial coefficient is zero when its upper integer is smaller than its nonnegative lower integer; the generalized polynomial convention for negative upper integers would give incorrect dimensions. For $n=0$, every twist on the single point ${\mathbf{P}}^0$ is trivial, so $h_0^0(m)=1$ for every $m$ and its higher cohomology is zero. This case supplies the base of dimension induction; the displayed top-cohomology formula below is for $n\ge 1$.

The standard $n+1$ affine opens have affine finite intersections. The acyclic cover theorem and affine vanishing identify sheaf cohomology with a Čech cochain complex ending in degree $n$. In particular $H^i({\mathbf{P}}^n,\mathcal{O}(m))=0$ for $i>n$.

Let $H=\{x_n=0\}\cong {\mathbf{P}}^{n-1}$, with inclusion $i$. Multiplication by $x_n$ gives the hyperplane exact sequence for twisting sheavesExactness is checked in the local trivializations, where a hyperplane equation is a non-zero-divisor and the quotient is its restriction to $H$. Since a closed immersion is an affine morphism, the first part identifies the cohomology of $i_{\ast }{\mathcal{O}}_H(m)$ with that on $H$.

For $n=1$, the long exact sequence in sheaf cohomology and $H\cong {\mathbf{P}}^0$ giveStarting from the given vanishing for sufficiently large $m$ and descending gives $h_1^1(m)=0$ for $m\ge -1$ and $h_1^1(m)=-m-1$ for $m\le -2$.

Now let $n\ge 2$, and assume the formulas in dimension $n-1$. Restrictionis surjective for every $m$: for $m\ge 0$ it is polynomial restriction, and for $m<0$ both groups are zero since $\dim H\ge 1$. The long exact sequence in sheaf cohomology therefore makes $H^1({\mathbf{P}}^n,\mathcal{O}(m-1))\to H^1({\mathbf{P}}^n,\mathcal{O}(m))$ injective. For $2\le j<n$, dimension induction gives $H^{j-1}(H,{\mathcal{O}}_H(m))=0$, so the same sequence makesinjective. Composing these injections up to a sufficiently large twist, where the target is zero by the given hypothesis, proves $H^j({\mathbf{P}}^n,\mathcal{O}(m))=0$ for every $m$ and $0<j<n$.

The remaining part of the long exact sequence in sheaf cohomology is consequentlywhere the last zero uses the affine-cover dimension bound on $H$. Hence $h_n^n(m-1)=h_n^n(m)+h_{n-1}^{n-1}(m)$. Using dimension induction and eventual vanishing, downward summation yieldsby telescoping Pascal's identity; for $m\ge -n$ it is zero. Thus, without invoking Serre duality in the induction,Finally, the canonical bundle of projective space is ${\omega }_{{\mathbf{P}}^n}\cong \mathcal{O}(-n-1)$, obtained by taking determinants in the dual Euler sequence. Serre duality therefore predictsIt pairs the two computed extreme-degree dimensions and pairs zero intermediate groups with zero intermediate groups, exactly as required.

For the closed immersion $i:Y\hookrightarrow X$, the ideal sheaf of a closed subscheme givesBecause the projective hypersurface is cut out by one homogeneous polynomial $f$ of degree $d$, multiplication by $f$ identifiesThis is the ideal-sheaf sequence of a projective hypersurface.

Choose a point $p\in C$ and project from $p$ to a line not containing $p$. Every line through $p$ meets the quartic in four points counted with intersection multiplicity by Bézout theorem; removing the fixed intersection at $p$ leaves a generic fibre of three points. This projection from a point on a plane curve therefore defines a nonconstant morphismof degree three, so the gonality is at most three.

It cannot be one, because a degree-one morphism between smooth projective curves is an isomorphism of algebraic varieties, whereas $C$ has genus three and the projective line has genus zero. It cannot be two either: that would make $C$ a hyperelliptic curve. The canonical map of a hyperelliptic curve is not a closed embedding, but for a smooth plane quartic the canonical map of a smooth plane curve is exactly its given embedding $C\hookrightarrow {\mathbb{P}}^2$, since $K_C\cong {\mathcal{O}}_C(1)$. This contradiction proves

Because ${\mathcal{A}}_{\bullet }$ is coherent and locally generated by ${\mathcal{A}}_1$, there is locally a finite-rank coherent sheaf $\mathcal{E}$ and a surjection of graded algebrasThe Relative Proj construction turns this into a closed immersionConsequently $Y\to X$ is a projective morphism and therefore a proper morphism. Since $X\to \mathrm{Spec}k$ is proper and proper morphisms are closed under composition, $Y\to \mathrm{Spec}k$ is proper.

The coincidence locus of two scheme morphisms is the fibre productThe diagonal morphism ${\Delta }_Y$ is a locally closed immersion, and this property is stable under base change of a morphism of schemes, so $E\hookrightarrow X$ is locally closed. The universal property of a fibre product says that a morphism $T\to X$ factors through $E$ exactly when $f{|}_T=g{|}_T$. Thus $E$ is the largest locally closed subscheme on which they coincide. If $Y$ is separated, ${\Delta }_Y$ is a closed immersion, and its base change $E\hookrightarrow X$ is closed.

The even-degree subring is the second Veronese subringwhere $u,v,w$ have degree one in the regraded ring. The canonical invariance of the Proj construction under passage to a Veronese subring givesOn homogeneous points this is the degree-two Veronese embeddingIt is an isomorphism onto the closed subschemeThus the quotient homomorphism $k[u,v,w]\twoheadrightarrow R^{(2)}$ supplies the requested closed immersion of $\mathrm{Proj}{R}^{(2)}$ into the projective plane.

An ${\mathcal{O}}_X$-module $\mathcal{F}$ is a quasi-coherent sheaf when every affine open $V=\mathrm{Spec}A\subseteq X$ has $\mathcal{F}{|}_V\cong \tilde{M}$ for some $A$-module $M$. For a morphism $f:X\to Y$, its direct image sheaf iswhile the pullback of a sheaf of modules is

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes, then on an affine open $V=\mathrm{Spec}A\subseteq X$ one has $Z\cap V=\mathrm{Spec}(A/I)$. The restriction of $i_{\ast }{\mathcal{O}}_Z$ corresponds to the cyclic $A$-module $A/I$, so it is finitely generated. Hence $i_{\ast }{\mathcal{O}}_Z$ is a coherent sheaf; more generally this is the direct image of a coherent sheaf under a closed immersion.

Coherence need not survive an arbitrary pushforward. Letbe the open immersion. The sheaf ${\mathcal{O}}_{D(x)}$ is coherent, butis not a finitely generated $k[x]$-module. Therefore $j_{\ast }{\mathcal{O}}_{D(x)}$ is not coherent.

Choose a finite affine cover $\mathcal{U}=(U_j)$ of the Noetherian scheme $X$. Because $X$ is separated, every finite intersection $U_J$ is affine. Its inverse image $V_J=Z\cap U_J$ under the closed immersion is also affine. By the definition of the direct image sheaf,The Čech complexes for $i_{\ast }\mathcal{F}$ on $\mathcal{U}$ and for $\mathcal{F}$ on the induced cover $(Z\cap U_j)$ are therefore identical, including their restriction maps. Both affine covers are acyclic for the relevant quasi-coherent sheaves, so the acyclic cover theorem givesfor every $q$. This is cohomology under a closed immersion.

For $X={\mathbb{P}}_k^n$, use its $n+1$ standard affine opens. The induced cover of $Z$ is still acyclic, and its Čech complex has no cochains in degrees greater than $n$. The cohomological dimension bound from an affine cover therefore yields

A morphism $f:X\to S$ is a separated morphism when its diagonal morphismis a closed immersion.

For the requested example, take the affine plane with doubled origin: glue two copies $U,V\cong {\mathbb{A}}_k^2$ by the identity onThe opens $U$ and $V$ are affine, while their intersection is the punctured affine plane, which is not affine. Indeed, its regular functions are still $k[x,y]$; if it were affine, the canonical map to $\mathrm{Spec}k[x,y]={\mathbb{A}}_k^2$ would be an isomorphism, contrary to the missing origin. The resulting scheme is not separated: in a separated scheme, the intersection of two affine opens is the inverse image of the closed diagonal inside their affine product and is therefore affine.

A closed immersion identifies its source homeomorphically with a closed subset and induces a surjection from the target structure sheaf to the pushed-forward source structure sheaf.

The quotient map $S\to S/I$ identifies $\mathrm{Proj}(S/I)$ with $V_{+}(I)\subseteq \mathrm{Proj}S$. On every standard open $D_{+}(f)$ it induces the surjectionwith kernel $(I_f{)}_0$. These affine-local maps glue, proving that the natural morphism $\mathrm{Proj}(S/I)\to \mathrm{Proj}S$ is a closed immersion.

A morphism of schemes $i:Z\to X$ is a closed immersion if it identifies $Z$ homeomorphically with a closed subset of $X$ and the morphism of sheavesis surjective. Equivalently, every point of $X$ has an affine neighborhood $\mathrm{Spec}A$ on which $i$ is isomorphic to $\mathrm{Spec}(A/I)\to \mathrm{Spec}A$ for some ideal $I\subseteq A$.

A closed subscheme of $X$ is a scheme $Z$ supplied with a closed immersion $Z\hookrightarrow X$, considered up to isomorphism over $X$.

On an affine open $W=\mathrm{Spec}A\subseteq X$, write $Z\cap W=V(I)$. Give $Z\cap W$ the reduced induced subscheme structure $\mathrm{Spec}(A/\sqrt{I})$. These constructions agree under localization and therefore glue to a reduced closed subscheme $Z\hookrightarrow X$.

Let $Z^{\prime }\hookrightarrow X$ be another closed immersion with the same underlying closed set. Affine-locally write $Z^{\prime }=\mathrm{Spec}(A/J)$. Since $V(J)=Z\cap W$, its vanishing ideal is $\sqrt{J}=\sqrt{I}$, and the inclusion $J\subseteq \sqrt{J}$ induces a quotient homomorphismContravariance of the spectrum of a commutative ring gives a factorizationThe quotient maps force these local factorizations to agree on overlaps, so they glue. They are also the only possible maps over $X$, which proves uniqueness.

Write $X=\mathrm{Spec}B$, $Y=\mathrm{Spec}A$, and let the morphism correspond to a homomorphism $\phi :A\to B$. Put $I=\ker \phi$ andThe injection $A/I\hookrightarrow B$ gives $X\to Z$, and the quotient $A\to A/I$ gives a closed immersion $Z\hookrightarrow Y$, so $f$ factors through $Z$.

If $f$ factors through another closed subscheme $Z^{\prime }=\mathrm{Spec}(A/J)$, then $J\subseteq \ker \phi =I$. The resulting quotient $A/J\to A/I$ induces the unique factorization $Z\to Z^{\prime }\to Y$. Thus $Z$ is the scheme-theoretic image.

It remains to identify its underlying set. A principal open subscheme $D(a)\subseteq \mathrm{Spec}A$ misses $f(X)$ exactly when $D(\phi (a))$ is empty, equivalently when $\phi (a)$ is a nilpotent element. Hence the ideal of functions vanishing set-theoretically on $f(X)$ has radical $\sqrt{\ker \phi }$. The closure is thereforeas required.

A morphism $X\to S$ is projective when it factors as a closed immersion $X\hookrightarrow {\mathbb{P}}_S^n$ followed by the projection to $S$. Every projective morphism is proper.