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The Abel map of an algebraic curve sends an effective Cartier divisor $D$ to its line bundle ${\mathcal{O}}_C(D)$. Its scheme fibre over a line bundle $L$ is the projective space of one-dimensional subspaces of $H^0(C,L)$. This follows in families: a relative effective Cartier divisor in the fixed class determines a nowhere fibre-zero section of $L$ up to a line bundle from the parameter space, equivalently a line subbundle of $H^0(C,L)\otimes {\mathcal{O}}_T$. Thus the scheme fibre is smooth, not merely reduced on its points. The dimensions can nevertheless jump, so the Abel map of an algebraic curve need not be a smooth morphism.

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.

For a degree-zero rational function regular on an open subset of projective space, each derivative $\partial f/\partial X_i$ is a regular homogeneous rational function of degree $-1$. These derivatives define a derivation with values in $\mathcal{O}(-1{)}^{\oplus (n+1)}$, hence a map from the Kähler differential sheaf. The last map is $(g_i)\mapsto {\sum }_i{X}_i{g}_i$. On $X_i\ne 0$, put $y_j=X_j/X_i$ and use frame $X_i^{-1}$. The derivative of $y_j$ has coordinate vector with $1$ at $j$, $-y_j$ at $i$ and zero elsewhere. Those vectors form a free basis of the kernel of $(b_i)\mapsto b_i+{\sum }_{j\ne i}{y}_j{b}_j$, proving exactness directly in every characteristic.

A cubic surface is a degree-three projective hypersurface in the projective space ${\mathbb{P}}^3$. Over an algebraically closed field every cubic surface contains a projective line. The smooth case has finitely many lines, although the incidence proof of existence does not require smoothness.

From a short exact sequence of sheaves $0\to K_{j+1}\to G_j\to K_j\to 0$, the long exact sequence in sheaf cohomology gives the displayed isomorphism whenever $H^q(X,G_j)=H^{q+1}(X,G_j)=0$. Repeating it along a locally free resolution shifts a fixed group to successively higher degrees. If all relevant kernel sheaves are quasi-coherent sheaves and there is a uniform vanishing bound for their sheaf cohomology, sufficiently many shifts force the original group to vanish. The cohomological dimension bound from an affine cover supplies such a bound for a closed subscheme of projective space.

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.

For an ample line bundle $L$ on a noetherian scheme $X$, every coherent sheaf $F$ becomes a globally generated sheaf after tensoring with $L^{\otimes n}$ for all sufficiently large $n$. Finitely many global sections suffice by finite global generation on a quasi-compact scheme. This is the global-generation characterization of an ample line bundle.^[ref]

For a closed subscheme of projective space, take $L={\iota }^{\ast }\mathcal{O}(1)$. One way to see why twisting helps is to choose finitely many module generators on each standard affine open subscheme. Multiplication by a sufficiently high power of its homogeneous coordinate extends each generator to a global section of a twist. Choose one exponent for all generators. On the original chart that coordinate is invertible, so these extended sections still generate; doing this for every chart proves global generation. Tensoring back by the inverse invertible sheaf gives a surjection from a finite direct sum of sheaves of negative twists onto $F$.

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.

A projective variety is a Zariski-closed subset of projective space ${\mathbb{P}}_k^n$, equivalently the common zero set of homogeneous polynomials. Homogeneity makes their vanishing independent of the nonzero scalar used to represent a projective point.

A morphism is a map locally given by regular coordinate functions in affine charts. Equivalently, near each point of a projective source embedded in ${\mathbb{P}}^n$, a morphism to ${\mathbb{P}}^m$ can be writtenusing homogeneous polynomials of a common degree that are not all zero on that neighbourhood. The expressions must agree on overlaps, and for a specified target variety their values must lie in it. A single global tuple is a sufficient description when available, but is not part of the general definition.

A quasi-projective variety is a locally closed subset of projective space: equivalently an open subset of some projective variety, with the induced algebraic variety structure. Affine varieties are examples, via the standard affine chart in projective space.

It is irreducible if it is nonempty and cannot be written as the union of two proper closed subsets. Equivalently, every two nonempty open subsets meet, or every nonempty open subset is dense. Closed and open are interpreted in the induced Zariski topology of $X$.

A regular function on $U$ is a function $f:U\to k$ locally representable, in an affine chart of the ambient projective space, as a quotient $a/b$ of polynomial functions with denominator nonzero on that neighbourhood. Equivalently it is locally a quotient of homogeneous forms of the same degree. The equal-degree condition makes the quotient independent of the chosen projective representative.

The ring ${\mathcal{O}}_{X,p}$ is the ring of germs of such functions near $p$. Two representatives $(U,f)$ and $(V,g)$ define the same germ when they agree on some smaller neighbourhood of $p$. Addition and multiplication use a common neighbourhood. It is a local ring: its unique maximal ideal consists of germs vanishing at $p$, and any germ nonzero at $p$ has a regular reciprocal on a smaller neighbourhood.

Nonzero scalar matrices form a subgroup $N$ of the general linear group, since products and inverses remain nonzero scalar matrices. Moreover, for every invertible $g$,Thus $N$ lies in the center of a group and is a normal subgroup. The quotient group is the complex projective linear group ${\mathrm{PGL}}_n(\mathbb{C})$, whose elements identify invertible matrices differing by a nonzero scalar.

The canonical group homomorphism $\pi :g\mapsto [g]$ obeys $[g][h]=[gh]$. Composing it with the given group representation givesConsequentlyThis is the projective representation associated with $R$: a nonzero scalar acts trivially on each line in the projective space of $V$. Its kernel of a group homomorphism consists of those $g$ for which $R(g)$ is scalar, which can be larger than the kernel of a group homomorphism of $R$ itself.