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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.

Over any field, the canonical bundle of projective space is $\mathcal{O}(-n-1)$. Indeed, dualizing the Euler sequence gives $0\to {\Omega }_{{\mathbf{P}}^n}^1\to \mathcal{O}(-1{)}^{\oplus (n+1)}\to \mathcal{O}\to 0$; taking determinants gives the result. Hence Serre duality identifies $H^i({\mathbf{P}}^n,\mathcal{O}(m))$ with the dual of $H^{n-i}({\mathbf{P}}^n,\mathcal{O}(-m-n-1))$.

On the standard charts of projective space, wedge the coordinate differentials in increasing order. The change $X_0/X_i=(X_i/X_0{)}^{-1}$ and $X_j/X_i=(X_j/X_0)/(X_i/X_0)$ gives the displayed Jacobian determinant. Multiplying the $i$th local generator by $(-1{)}^i$ removes this sign and yields the transition functions of the twisting sheaf on projective space with degree $-n-1$.

Over complex projective space, its fibre at a line $\ell \subseteq {\mathbb{C}}^{n+1}$ is $\ell$ itself. It is a holomorphic line bundle contained in the trivial bundle. Its dual is the hyperplane line bundle.

A globally generated line bundle on a quasi-compact scheme is generated by finitely many global sections. Each point has a section generating on an open neighbourhood, and quasi-compactness gives a finite subcover. This supplies a map into a fixed finite-dimensional projective space associated with that line bundle.

The embedding line bundle has at least four independent global sections. If it is nonspecial, the Riemann-Roch theorem gives $h^0=6-g$, so $g\le 2$. If it is special, the Clifford inequality for curves gives $h^0\le 3$, a contradiction. The smooth projective curve need not span its full ambient projective space.

The only regular functions defined on all of a nonempty projective space over a field $k$ are the constants:For ${\mathbb{P}}^1$, this also follows by gluing $k[t]$ and $k[t^{-1}]$ inside $k[t,t^{-1}]$: their intersection is $k$.

A hyperplane divisor on projective space is the effective Cartier divisor cut out by one nonzero linear homogeneous polynomial. Its class generates $\mathrm{Cl}({\mathbb{P}}_k^n)\cong \mathbb{Z}$.

For $n\ge 1$, the top Čech cohomology of the standard affine cover of projective space is the degree-$m$ quotient of the Laurent ring by the sum of rings in which at least one variable has not been inverted. Its basis consists of Laurent monomials $X_0^{a_0}\cdots X_n^{a_n}$ with every $a_i<0$ and ${\sum }_i{a}_i=m$. Hence it vanishes for $m\ge -n$ and has dimension $\left(\genfrac{}{}{0ex}{}{-m-1}{n}\right)$ for $m\le -n-1$.

For a finite generating space $V\subseteq H^0(X,\mathcal{L})$ on a projective scheme, its map to projective space is a closed immersion exactly when, after algebraic closure of the ground field, the displayed evaluation is surjective for every length-two closed subscheme $E$. Two distinct points test point separation; a double point tests a direction in the Zariski tangent space. For the complete section space, this characterizes a very ample line bundle.

A line bundle on a scheme is a locally free sheaf of rank one. Its global sections can define a morphism to projective space when they have no common zero.