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The projective morphism ${\mathrm{Proj}}_X{⨁}_{j\ge 0}{\mathcal{I}}^j\to X$ associated to the powers of the ideal sheaf of a closed centre. Off that centre it is an isomorphism. Blowing up a smooth point of an $n$-dimensional smooth variety replaces the point by a projective space ${\mathbb{P}}^{n-1}$, the algebraic exceptional divisor.

We work over an algebraically closed field $k$, so the assertion concerns a geometric projective line. Let $V=k^4$ and let $W$ be the vector space of homogeneous cubic forms on $V$. It has dimension $\left(\genfrac{}{}{0ex}{}{6}{3}\right)=20$, so the parameter space of cubic equations up to nonzero scalar is $\mathbb{P}(W)={\mathbb{P}}^{19}$. The Grassmannian $G=\mathrm{Gr}(2,V)$ parametrizes lines in ${\mathbb{P}}^3$. Its standard graph charts are four-dimensional affine spaces, so its dimension is four. The Plücker embedding realizes it as a closed quadric in ${\mathbb{P}}^5$, proving projectivity. The general linear group ${\mathrm{GL}}_4$ acts transitively on two-planes and is an irreducible open subset of the affine space of matrices; its orbit map proves that $G$ is irreducible.

Consider the incidence varietyRestriction of a cubic to a fixed line is surjective onto the four-dimensional space of binary cubic forms: after choosing coordinates with that line given by $x_2=x_3=0$, extend any binary cubic using $x_0,x_1$ alone. Thus its kernel has dimension sixteen. On each Grassmannian chart these restriction maps depend algebraically on the line parameters and have constant rank four. Their kernels form a rank-sixteen vector bundle, and $\mathcal{I}$ is its projective bundle, with fibre ${\mathbb{P}}^{15}$. In particular, $\mathcal{I}$ is irreducible of dimension $4+15=19$. It is closed in the product, since restriction vanishes precisely when four polynomial coefficients vanish.

The projection $\pi :\mathcal{I}\to \mathbb{P}(W)$ is a projective morphism, hence its image is closed. It remains to prove that this image has dimension nineteen. Equal dimensions alone would not suffice: we must control a fibre locally.

TakeIn the open Grassmannian chart around ${\ell }_0$, every line is uniquely a graphRestriction of $F_0$ to that line isThe four monomials are linearly independent over every field. Hence the only line in this chart lying on $F_0$ has $a_0=a_1=b_0=b_1=0$. The fibre ${\pi }^{-1}([F_0])$ therefore has local dimension zero at $({\ell }_0,[F_0])$. We do not need $F_0$ to define a smooth surface, nor do we claim its entire fibre is finite.

The fiber dimension theorem, in its local dimension inequality, givesThus the closed image has dimension at least nineteen. A proper closed subset of the irreducible ${\mathbb{P}}^{19}$ has smaller dimension, so $\pi (\mathcal{I})={\mathbb{P}}^{19}$. Every cubic equation consequently has a line in its zero set. In particular, every nonsingular cubic surface contains a line. The proof in fact establishes existence for arbitrary cubic surfaces over an algebraically closed field; it does not assert a rational line over an arbitrary nonclosed ground field.

Let $f:{\mathbb{P}}_k^1\to \mathrm{Spec}k$ be the structure map, a projective morphism. Let $D$ be the union of the two rational points $[0:1]$ and $[1:0]$. The homogeneous polynomial $t_0{t}_1$ defines this effective Cartier divisor and gives the exact sequence of coherent sheavesThe twisting sheaf on projective space $\mathcal{O}(-2)$ has no nonzero global sections: the chart calculation of Question 3(iv) now requires $a(t)=t^{-2}b(t^{-1})$, again impossible for nonzero polynomials. We also have $\Gamma ({\mathbb{P}}^1,\mathcal{O})=k$ and $\Gamma (D,{\mathcal{O}}_D)=k\oplus k$.

The direct image sheaf under $f$ is the module of global sections on the one-point scheme. Consequently the induced sequence isIts last nonzero map has cokernel isomorphic to $k$, since $(a,b)\mapsto b-a$ has the diagonal as its kernel. Thus the induced sequence is not exact at $f_{\ast }{\mathcal{O}}_D$, proving that projective direct image need not preserve surjections.

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.

A projective scheme over a base $S$ is an $S$-scheme admitting a closed immersion into some projective space ${\mathbb{P}}_S^n$. Every projective morphism is proper.

Projectivity is a property of the scheme over its specified base, expressed by its structural projective morphism.