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Projective variety

Inalgebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space ${\displaystyle \mathbb {P} ^{n}}$ over k that is the zero-locus of some finite family of homogeneous polynomialsofn + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvarietyof${\displaystyle \mathbb {P} ^{n}}$.

A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial.

IfX is a projective variety defined by a homogeneous prime ideal I, then the quotient ring

${\displaystyle k[x_{0},\ldots ,x_{n}]/I}$

is called the homogeneous coordinate ringofX. Basic invariants of X such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring.

Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, but Chow's lemma describes the close relation of these two notions. Showing that a variety is projective is done by studying line bundlesordivisorsonX.

A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties, Serre duality can be viewed as an analog of Poincaré duality. It also leads to the Riemann–Roch theorem for projective curves, i.e., projective varieties of dimension 1. The theory of projective curves is particularly rich, including a classification by the genus of the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties.^[1] Hilbert schemes parametrize closed subschemes of ${\displaystyle \mathbb {P} ^{n}}$ with prescribed Hilbert polynomial. Hilbert schemes, of which Grassmannians are special cases, are also projective schemes in their own right. Geometric invariant theory offers another approach. The classical approaches include the Teichmüller space and Chow varieties.

A particularly rich theory, reaching back to the classics, is available for complex projective varieties, i.e., when the polynomials defining X have complex coefficients. Broadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. For example, the theory of holomorphic vector bundles (more generally coherent analytic sheaves) on X coincide with that of algebraic vector bundles. Chow's theorem says that a subset of projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. The combination of analytic and algebraic methods for complex projective varieties lead to areas such as Hodge theory.

Variety and scheme structure[edit]

Variety structure[edit]

Let k be an algebraically closed field. The basis of the definition of projective varieties is projective space ${\displaystyle \mathbb {P} ^{n}}$, which can be defined in different, but equivalent ways:

• as the set of all lines through the origin in ${\displaystyle k^{n+1}}$ (i.e., all one-dimensional vector subspaces of ${\displaystyle k^{n+1}}$)
• as the set of tuples ${\displaystyle (x_{0},\dots ,x_{n})\in k^{n+1}}$, with ${\displaystyle x_{0},\dots ,x_{n}}$ not all zero, modulo the equivalence relation $${\displaystyle (x_{0},\dots ,x_{n})\sim \lambda (x_{0},\dots ,x_{n})}$$ for any ${\displaystyle \lambda \in k\setminus \{0\}}$. The equivalence class of such a tuple is denoted by $${\displaystyle [x_{0}:\dots :x_{n}].}$$ This equivalence class is the general point of projective space. The numbers ${\displaystyle x_{0},\dots ,x_{n}}$ are referred to as the homogeneous coordinates of the point.

Aprojective variety is, by definition, a closed subvariety of ${\displaystyle \mathbb {P} ^{n}}$, where closed refers to the Zariski topology.^[2] In general, closed subsets of the Zariski topology are defined to be the common zero-locus of a finite collection of homogeneous polynomial functions. Given a polynomial ${\displaystyle f\in k[x_{0},\dots ,x_{n}]}$, the condition

${\displaystyle f([x_{0}:\dots :x_{n}])=0}$

does not make sense for arbitrary polynomials, but only if fishomogeneous, i.e., the degrees of all the monomials (whose sum is f) are the same. In this case, the vanishing of

${\displaystyle f(\lambda x_{0},\dots ,\lambda x_{n})=\lambda ^{\deg f}f(x_{0},\dots ,x_{n})}$

is independent of the choice of ${\displaystyle \lambda \neq 0}$.

Therefore, projective varieties arise from homogeneous prime ideals Iof${\displaystyle k[x_{0},\dots ,x_{n}]}$, and setting

${\displaystyle X=\left\{[x_{0}:\dots :x_{n}]\in \mathbb {P} ^{n},f([x_{0}:\dots :x_{n}])=0{\text{ for all }}f\in I\right\}.}$

Moreover, the projective variety X is an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation axiom. Thus, the local study of X (e.g., singularity) reduces to that of an affine variety. The explicit structure is as follows. The projective space ${\displaystyle \mathbb {P} ^{n}}$ is covered by the standard open affine charts

${\displaystyle U_{i}=\{[x_{0}:\dots :x_{n}],x_{i}\neq 0\},}$

which themselves are affine n-spaces with the coordinate ring

${\displaystyle k\left[y_{1}^{($i)},\dots ,y_{n}^{(i)}\right],\quad y_{j}^{(i)}=x_{j}/x_{i}.}

Say i = 0 for the notational simplicity and drop the superscript (0). Then ${\displaystyle X\cap U_{0}}$ is a closed subvariety of ${\displaystyle U_{0}\simeq \mathbb {A} ^{n}}$ defined by the ideal of ${\displaystyle k[y_{1},\dots ,y_{n}]}$ generated by

${\displaystyle f(1,y_{1},\dots ,y_{n})}$

for all finI. Thus, X is an algebraic variety covered by (n+1) open affine charts ${\displaystyle X\cap U_{i}}$.

Note that X is the closure of the affine variety ${\displaystyle X\cap U_{0}}$in${\displaystyle \mathbb {P} ^{n}}$. Conversely, starting from some closed (affine) variety ${\displaystyle V\subset U_{0}\simeq \mathbb {A} ^{n}}$, the closure of Vin${\displaystyle \mathbb {P} ^{n}}$ is the projective variety called the projective completionofV. If ${\displaystyle I\subset k[y_{1},\dots ,y_{n}]}$ defines V, then the defining ideal of this closure is the homogeneous ideal^[3]of${\displaystyle k[x_{0},\dots ,x_{n}]}$ generated by

${\displaystyle x_{0}^{\deg($f)}f(x_{1}/x_{0},\dots ,x_{n}/x_{0})}

for all finI.

For example, if V is an affine curve given by, say, ${\displaystyle y^{2}=x^{3}+ax+b}$ in the affine plane, then its projective completion in the projective plane is given by ${\displaystyle y^{2}z=x^{3}+axz^{2}+bz^{3}.}$

Projective schemes[edit]

For various applications, it is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e., ${\displaystyle \mathbb {P} ^{n}($k)} is a scheme which it is a union of (n + 1) copies of the affine n-space kⁿ. More generally,^[4] projective space over a ring A is the union of the affine schemes

${\displaystyle U_{i}=\operatorname {Spec} A[x_{0}/x_{i},\dots ,x_{n}/x_{i}],\quad 0\leq i\leq n,}$

in such a way the variables match up as expected. The set of closed pointsof${\displaystyle \mathbb {P} _{k}^{n}}$, for algebraically closed fields k, is then the projective space ${\displaystyle \mathbb {P} ^{n}($k)} in the usual sense.

An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted "Spec", which defines an affine scheme.^[5] For example, if A is a ring, then

${\displaystyle \mathbb {P} _{A}^{n}=\operatorname {Proj} A[x_{0},\ldots ,x_{n}].}$

IfR is a quotientof${\displaystyle k[x_{0},\ldots ,x_{n}]}$ by a homogeneous ideal I, then the canonical surjection induces the closed immersion

${\displaystyle \operatorname {Proj} R\hookrightarrow \mathbb {P} _{k}^{n}.}$

Compared to projective varieties, the condition that the ideal I be a prime ideal was dropped. This leads to a much more flexible notion: on the one hand the topological space ${\displaystyle X=\operatorname {Proj} R}$ may have multiple irreducible components. Moreover, there may be nilpotent functions on X.

Closed subschemes of ${\displaystyle \mathbb {P} _{k}^{n}}$ correspond bijectively to the homogeneous ideals Iof${\displaystyle k[x_{0},\ldots ,x_{n}]}$ that are saturated; i.e., ${\displaystyle I:(x_{0},\dots ,x_{n})=I.}$^[6] This fact may be considered as a refined version of projective Nullstellensatz.

We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector space V over k, we let

${\displaystyle \mathbb {P} ($V)=\operatorname {Proj} k[V]}

where ${\displaystyle k[$V]=\operatorname {Sym} (V^{*})} is the symmetric algebraof${\displaystyle V^{*}}$.^[7] It is the projectivizationofV; i.e., it parametrizes lines in V. There is a canonical surjective map ${\displaystyle \pi :V\setminus \{0\}\to \mathbb {P} ($V)} , which is defined using the chart described above.^[8] One important use of the construction is this (cf., § Duality and linear system). A divisor D on a projective variety X corresponds to a line bundle L. One then set

${\displaystyle |D|=\mathbb {P} (\Gamma (X,L))}$;

it is called the complete linear systemofD.

Projective space over any scheme S can be defined as a fiber product of schemes

${\displaystyle \mathbb {P} _{S}^{n}=\mathbb {P} _{\mathbb {Z} }^{n}\times _{\operatorname {Spec} \mathbb {Z} }S.}$

If${\displaystyle {\mathcal {O}}($1)} is the twisting sheaf of Serreon${\displaystyle \mathbb {P} _{\mathbb {Z} }^{n}}$, we let ${\displaystyle {\mathcal {O}}($1)} denote the pullbackof${\displaystyle {\mathcal {O}}($1)} to${\displaystyle \mathbb {P} _{S}^{n}}$; that is, ${\displaystyle {\mathcal {O}}($1)=g^{*}({\mathcal {O}}(1))} for the canonical map ${\displaystyle g:\mathbb {P} _{S}^{n}\to \mathbb {P} _{\mathbb {Z} }^{n}.}$

A scheme X → S is called projective over S if it factors as a closed immersion

${\displaystyle X\to \mathbb {P} _{S}^{n}}$

followed by the projection to S.

A line bundle (or invertible sheaf) ${\displaystyle {\mathcal {L}}}$ on a scheme X over S is said to be very ample relative to S if there is an immersion (i.e., an open immersion followed by a closed immersion)

${\displaystyle i:X\to \mathbb {P} _{S}^{n}}$

for some n so that ${\displaystyle {\mathcal {O}}($1)} pullbacks to ${\displaystyle {\mathcal {L}}}$. Then a S-scheme X is projective if and only if it is proper and there exists a very ample sheaf on X relative to S. Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed. Conversely, if X is projective, then the pullback of ${\displaystyle {\mathcal {O}}($1)} under the closed immersion of X into a projective space is very ample. That "projective" implies "proper" is deeper: the main theorem of elimination theory.

Relation to complete varieties[edit]

By definition, a variety is complete, if it is proper over k. The valuative criterion of properness expresses the intuition that in a proper variety, there are no points "missing".

There is a close relation between complete and projective varieties: on the one hand, projective space and therefore any projective variety is complete. The converse is not true in general. However:

• Asmooth curve C is projective if and only if it is complete. This is proved by identifying C with the set of discrete valuation rings of the function field k(C) over k. This set has a natural Zariski topology called the Zariski–Riemann space.
• Chow's lemma states that for any complete variety X, there is a projective variety Z and a birational morphism Z → X.^[9] (Moreover, through normalization, one can assume this projective variety is normal.)

Some properties of a projective variety follow from completeness. For example,

${\displaystyle \Gamma (X,{\mathcal {O}}_{X})=k}$

for any projective variety X over k.^[10] This fact is an algebraic analogue of Liouville's theorem (any holomorphic function on a connected compact complex manifold is constant). In fact, the similarity between complex analytic geometry and algebraic geometry on complex projective varieties goes much further than this, as is explained below.

Quasi-projective varieties are, by definition, those which are open subvarieties of projective varieties. This class of varieties includes affine varieties. Affine varieties are almost never complete (or projective). In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globally regular functions on a projective variety.

Examples and basic invariants[edit]

By definition, any homogeneous ideal in a polynomial ring yields a projective scheme (required to be prime ideal to give a variety). In this sense, examples of projective varieties abound. The following list mentions various classes of projective varieties which are noteworthy since they have been studied particularly intensely. The important class of complex projective varieties, i.e., the case ${\displaystyle k=\mathbb {C} }$, is discussed further below.

The product of two projective spaces is projective. In fact, there is the explicit immersion (called Segre embedding)

${\displaystyle {\begin{cases}\mathbb {P} ^{n}\times \mathbb {P} ^{m}\to \mathbb {P} ^{(n+1)(m+1)-1}\\(x_{i},y_{j})\mapsto x_{i}y_{j}\end{cases}}}$

As a consequence, the product of projective varieties over k is again projective. The Plücker embedding exhibits a Grassmannian as a projective variety. Flag varieties such as the quotient of the general linear group ${\displaystyle \mathrm {GL} _{n}($k)} modulo the subgroup of upper triangular matrices, are also projective, which is an important fact in the theory of algebraic groups.^[11]

Homogeneous coordinate ring and Hilbert polynomial[edit]

As the prime ideal P defining a projective variety X is homogeneous, the homogeneous coordinate ring

${\displaystyle R=k[x_{0},\dots ,x_{n}]/P}$

is a graded ring, i.e., can be expressed as the direct sum of its graded components:

${\displaystyle R=\bigoplus _{n\in \mathbb {N} }R_{n}.}$

There exists a polynomial P such that ${\displaystyle \dim R_{n}=P($n)} for all sufficiently large n; it is called the Hilbert polynomialofX. It is a numerical invariant encoding some extrinsic geometry of X. The degree of P is the dimension rofX and its leading coefficient times r! is the degree of the variety X. The arithmetic genusofX is (−1)^r (P(0) − 1) when X is smooth.

For example, the homogeneous coordinate ring of ${\displaystyle \mathbb {P} ^{n}}$is${\displaystyle k[x_{0},\ldots ,x_{n}]}$ and its Hilbert polynomial is ${\displaystyle P($z)={\binom {z+n}{n}}} ; its arithmetic genus is zero.

If the homogeneous coordinate ring R is an integrally closed domain, then the projective variety X is said to be projectively normal. Note, unlike normality, projective normality depends on R, the embedding of X into a projective space. The normalization of a projective variety is projective; in fact, it's the Proj of the integral closure of some homogeneous coordinate ring of X.

Degree[edit]

Let ${\displaystyle X\subset \mathbb {P} ^{N}}$ be a projective variety. There are at least two equivalent ways to define the degree of X relative to its embedding. The first way is to define it as the cardinality of the finite set

${\displaystyle \#(X\cap H_{1}\cap \cdots \cap H_{d})}$

where d is the dimension of X and H_i's are hyperplanes in "general positions". This definition corresponds to an intuitive idea of a degree. Indeed, if X is a hypersurface, then the degree of X is the degree of the homogeneous polynomial defining X. The "general positions" can be made precise, for example, by intersection theory; one requires that the intersection is proper and that the multiplicities of irreducible components are all one.

The other definition, which is mentioned in the previous section, is that the degree of X is the leading coefficient of the Hilbert polynomialofX times (dim X)!. Geometrically, this definition means that the degree of X is the multiplicity of the vertex of the affine cone over X.^[12]

Let ${\displaystyle V_{1},\dots ,V_{r}\subset \mathbb {P} ^{N}}$ be closed subschemes of pure dimensions that intersect properly (they are in general position). If m_i denotes the multiplicity of an irreducible component Z_i in the intersection (i.e., intersection multiplicity), then the generalization of Bézout's theorem says:^[13]

${\displaystyle \sum _{1}^{s}m_{i}\deg Z_{i}=\prod _{1}^{r}\deg V_{i}.}$

The intersection multiplicity m_i can be defined as the coefficient of Z_i in the intersection product ${\displaystyle V_{1}\cdot \cdots \cdot V_{r}}$ in the Chow ringof${\displaystyle \mathbb {P} ^{N}}$.

In particular, if ${\displaystyle H\subset \mathbb {P} ^{N}}$ is a hypersurface not containing X, then

${\displaystyle \sum _{1}^{s}m_{i}\deg Z_{i}=\deg($X)\deg(H)}

where Z_i are the irreducible components of the scheme-theoretic intersectionofX and H with multiplicity (length of the local ring) m_i.

A complex projective variety can be viewed as a compact complex manifold; the degree of the variety (relative to the embedding) is then the volume of the variety as a manifold with respect to the metric inherited from the ambient complex projective space. A complex projective variety can be characterized as a minimizer of the volume (in a sense).

The ring of sections[edit]

Let X be a projective variety and L a line bundle on it. Then the graded ring

${\displaystyle R(X,L)=\bigoplus _{n=0}^{\infty }H^{0}(X,L^{\otimes n})}$

is called the ring of sectionsofL. If Lisample, then Proj of this ring is X. Moreover, if X is normal and L is very ample, then ${\displaystyle R(X,L)}$ is the integral closure of the homogeneous coordinate ring of X determined by L; i.e., ${\displaystyle X\hookrightarrow \mathbb {P} ^{N}}$ so that ${\displaystyle {\mathcal {O}}_{\mathbb {P} ^{N}}($1)} pulls-back to L.^[14]

For applications, it is useful to allow for divisors (or${\displaystyle \mathbb {Q} }$-divisors) not just line bundles; assuming X is normal, the resulting ring is then called a generalized ring of sections. If ${\displaystyle K_{X}}$ is a canonical divisoronX, then the generalized ring of sections

${\displaystyle R(X,K_{X})}$

is called the canonical ringofX. If the canonical ring is finitely generated, then Proj of the ring is called the canonical modelofX. The canonical ring or model can then be used to define the Kodaira dimensionofX.

Projective curves[edit]

Projective schemes of dimension one are called projective curves. Much of the theory of projective curves is about smooth projective curves, since the singularities of curves can be resolved by normalization, which consists in taking locally the integral closure of the ring of regular functions. Smooth projective curves are isomorphic if and only if their function fields are isomorphic. The study of finite extensions of

${\displaystyle \mathbb {F} _{p}($t),}

or equivalently smooth projective curves over ${\displaystyle \mathbb {F} _{p}}$ is an important branch in algebraic number theory.^[15]

A smooth projective curve of genus one is called an elliptic curve. As a consequence of the Riemann–Roch theorem, such a curve can be embedded as a closed subvariety in ${\displaystyle \mathbb {P} ^{2}}$. In general, any (smooth) projective curve can be embedded in ${\displaystyle \mathbb {P} ^{3}}$ (for a proof, see Secant variety#Examples). Conversely, any smooth closed curve in ${\displaystyle \mathbb {P} ^{2}}$ of degree three has genus one by the genus formula and is thus an elliptic curve.

A smooth complete curve of genus greater than or equal to two is called a hyperelliptic curve if there is a finite morphism ${\displaystyle C\to \mathbb {P} ^{1}}$ of degree two.^[16]

Projective hypersurfaces[edit]

Every irreducible closed subset of ${\displaystyle \mathbb {P} ^{n}}$ of codimension one is a hypersurface; i.e., the zero set of some homogeneous irreducible polynomial.^[17]

Abelian varieties[edit]

Another important invariant of a projective variety X is the Picard group ${\displaystyle \operatorname {Pic} ($X)} ofX, the set of isomorphism classes of line bundles on X. It is isomorphic to ${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})}$ and therefore an intrinsic notion (independent of embedding). For example, the Picard group of ${\displaystyle \mathbb {P} ^{n}}$ is isomorphic to ${\displaystyle \mathbb {Z} }$ via the degree map. The kernel of ${\displaystyle \deg :\operatorname {Pic} ($X)\to \mathbb {Z} } is not only an abstract abelian group, but there is a variety called the Jacobian varietyofX, Jac(X), whose points equal this group. The Jacobian of a (smooth) curve plays an important role in the study of the curve. For example, the Jacobian of an elliptic curve EisE itself. For a curve X of genus g, Jac(X) has dimension g.

Varieties, such as the Jacobian variety, which are complete and have a group structure are known as abelian varieties, in honor of Niels Abel. In marked contrast to affine algebraic groups such as ${\displaystyle GL_{n}($k)} , such groups are always commutative, whence the name. Moreover, they admit an ample line bundle and are thus projective. On the other hand, an abelian scheme may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties and K3 surfaces.

Projections[edit]

Let ${\displaystyle E\subset \mathbb {P} ^{n}}$ be a linear subspace; i.e., ${\displaystyle E=\{s_{0}=s_{1}=\cdots =s_{r}=0\}}$ for some linearly independent linear functionals s_i. Then the projection from E is the (well-defined) morphism

${\displaystyle {\begin{cases}\phi :\mathbb {P} ^{n}-E\to \mathbb {P} ^{r}\\x\mapsto [s_{0}($x):\cdots :s_{r}(x)]\end{cases}}}

The geometric description of this map is as follows:^[18]

• We view ${\displaystyle \mathbb {P} ^{r}\subset \mathbb {P} ^{n}}$ so that it is disjoint from E. Then, for any ${\displaystyle x\in \mathbb {P} ^{n}\setminus E}$, $${\displaystyle \phi ($$x)=W_{x}\cap \mathbb {P} ^{r},} where ${\displaystyle W_{x}}$ denotes the smallest linear space containing E and x (called the joinofE and x.)
• ${\displaystyle \phi ^{-1}(\{y_{i}\neq 0\})=\{s_{i}\neq 0\},}$ where ${\displaystyle y_{i}}$ are the homogeneous coordinates on ${\displaystyle \mathbb {P} ^{r}.}$
• For any closed subscheme ${\displaystyle Z\subset \mathbb {P} ^{n}}$ disjoint from E, the restriction ${\displaystyle \phi :Z\to \mathbb {P} ^{r}}$ is a finite morphism.^[19]

Projections can be used to cut down the dimension in which a projective variety is embedded, up to finite morphisms. Start with some projective variety ${\displaystyle X\subset \mathbb {P} ^{n}.}$If${\displaystyle n>\dim X,}$ the projection from a point not on X gives ${\displaystyle \phi :X\to \mathbb {P} ^{n-1}.}$ Moreover, ${\displaystyle \phi }$ is a finite map to its image. Thus, iterating the procedure, one sees there is a finite map

${\displaystyle X\to \mathbb {P} ^{d},\quad d=\dim X.}$

This result is the projective analog of Noether's normalization lemma. (In fact, it yields a geometric proof of the normalization lemma.)

The same procedure can be used to show the following slightly more precise result: given a projective variety X over a perfect field, there is a finite birational morphism from X to a hypersurface Hin${\displaystyle \mathbb {P} ^{d+1}.}$^[20] In particular, if X is normal, then it is the normalization of H.

Duality and linear system[edit]

While a projective n-space ${\displaystyle \mathbb {P} ^{n}}$ parameterizes the lines in an affine n-space, the dual of it parametrizes the hyperplanes on the projective space, as follows. Fix a field k. By ${\displaystyle {\breve {\mathbb {P} }}_{k}^{n}}$, we mean a projective n-space

${\displaystyle {\breve {\mathbb {P} }}_{k}^{n}=\operatorname {Proj} (k[u_{0},\dots ,u_{n}])}$

equipped with the construction:

${\displaystyle f\mapsto H_{f}=\{\alpha _{0}x_{0}+\cdots +\alpha _{n}x_{n}=0\}}$, a hyperplane on ${\displaystyle \mathbb {P} _{L}^{n}}$

where ${\displaystyle f:\operatorname {Spec} L\to {\breve {\mathbb {P} }}_{k}^{n}}$ is an L-pointof${\displaystyle {\breve {\mathbb {P} }}_{k}^{n}}$ for a field extension Lofk and ${\displaystyle \alpha _{i}=f^{*}(u_{i})\in L.}$

For each L, the construction is a bijection between the set of L-points of ${\displaystyle {\breve {\mathbb {P} }}_{k}^{n}}$ and the set of hyperplanes on ${\displaystyle \mathbb {P} _{L}^{n}}$. Because of this, the dual projective space ${\displaystyle {\breve {\mathbb {P} }}_{k}^{n}}$ is said to be the moduli space of hyperplanes on ${\displaystyle \mathbb {P} _{k}^{n}}$.

A line in ${\displaystyle {\breve {\mathbb {P} }}_{k}^{n}}$ is called a pencil: it is a family of hyperplanes on ${\displaystyle \mathbb {P} _{k}^{n}}$ parametrized by ${\displaystyle \mathbb {P} _{k}^{1}}$.

IfV is a finite-dimensional vector space over k, then, for the same reason as above, ${\displaystyle \mathbb {P} (V^{*})=\operatorname {Proj} (\operatorname {Sym} ($V))} is the space of hyperplanes on ${\displaystyle \mathbb {P} ($V)} . An important case is when V consists of sections of a line bundle. Namely, let X be an algebraic variety, L a line bundle on X and ${\displaystyle V\subset \Gamma (X,L)}$ a vector subspace of finite positive dimension. Then there is a map:^[21]

${\displaystyle {\begin{cases}\varphi _{V}:X\setminus B\to \mathbb {P} (V^{*})\\x\mapsto H_{x}=\{s\in V|s($x)=0\}\end{cases}}}

determined by the linear system V, where B, called the base locus, is the intersection of the divisors of zero of nonzero sections in V (see Linear system of divisors#A map determined by a linear system for the construction of the map).

Cohomology of coherent sheaves[edit]

Let X be a projective scheme over a field (or, more generally over a Noetherian ring A). Cohomology of coherent sheaves ${\displaystyle {\mathcal {F}}}$onX satisfies the following important theorems due to Serre:

1. ${\displaystyle H^{p}(X,{\mathcal {F}})}$ is a finite-dimensional k-vector space for any p.
2. There exists an integer ${\displaystyle n_{0}}$ (depending on ${\displaystyle {\mathcal {F}}}$; see also Castelnuovo–Mumford regularity) such that $${\displaystyle H^{p}(X,{\mathcal {F}}($$n))=0} for all ${\displaystyle n\geq n_{0}}$ and p > 0, where ${\displaystyle {\mathcal {F}}($n)={\mathcal {F}}\otimes {\mathcal {O}}(n)} is the twisting with a power of a very ample line bundle ${\displaystyle {\mathcal {O}}($1).}

These results are proven reducing to the case ${\displaystyle X=\mathbb {P} ^{n}}$ using the isomorphism

${\displaystyle H^{p}(X,{\mathcal {F}})=H^{p}(\mathbb {P} ^{r},{\mathcal {F}}),p\geq 0}$

where in the right-hand side ${\displaystyle {\mathcal {F}}}$ is viewed as a sheaf on the projective space by extension by zero.^[22] The result then follows by a direct computation for ${\displaystyle {\mathcal {F}}={\mathcal {O}}_{\mathbb {P} ^{r}}($n),} n any integer, and for arbitrary ${\displaystyle {\mathcal {F}}}$ reduces to this case without much difficulty.^[23]

As a corollary to 1. above, if f is a projective morphism from a noetherian scheme to a noetherian ring, then the higher direct image ${\displaystyle R^{p}f_{*}{\mathcal {F}}}$ is coherent. The same result holds for proper morphisms f, as can be shown with the aid of Chow's lemma.

Sheaf cohomology groups Hⁱ on a noetherian topological space vanish for i strictly greater than the dimension of the space. Thus the quantity, called the Euler characteristicof${\displaystyle {\mathcal {F}}}$,

${\displaystyle \chi ({\mathcal {F}})=\sum _{i=0}^{\infty }(-1)^{i}\dim H^{i}(X,{\mathcal {F}})}$

is a well-defined integer (for X projective). One can then show ${\displaystyle \chi ({\mathcal {F}}($n))=P(n)} for some polynomial P over rational numbers.^[24] Applying this procedure to the structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$, one recovers the Hilbert polynomial of X. In particular, if X is irreducible and has dimension r, the arithmetic genus of X is given by

${\displaystyle (-1)^{r}(\chi ({\mathcal {O}}_{X})-1),}$

which is manifestly intrinsic; i.e., independent of the embedding.

The arithmetic genus of a hypersurface of degree dis${\displaystyle {\binom {d-1}{n}}}$in${\displaystyle \mathbb {P} ^{n}}$. In particular, a smooth curve of degree din${\displaystyle \mathbb {P} ^{2}}$ has arithmetic genus ${\displaystyle (d-1)(d-2)/2}$. This is the genus formula.

Smooth projective varieties[edit]

Let X be a smooth projective variety where all of its irreducible components have dimension n. In this situation, the canonical sheaf ω_X, defined as the sheaf of Kähler differentials of top degree (i.e., algebraic n-forms), is a line bundle.

Serre duality[edit]

Serre duality states that for any locally free sheaf ${\displaystyle {\mathcal {F}}}$onX,

${\displaystyle H^{i}(X,{\mathcal {F}})\simeq H^{n-i}(X,{\mathcal {F}}^{\vee }\otimes \omega _{X})'}$

where the superscript prime refers to the dual space and ${\displaystyle {\mathcal {F}}^{\vee }}$ is the dual sheaf of ${\displaystyle {\mathcal {F}}}$. A generalization to projective, but not necessarily smooth schemes is known as Verdier duality.

Riemann–Roch theorem[edit]

For a (smooth projective) curve X, H² and higher vanish for dimensional reason and the space of the global sections of the structure sheaf is one-dimensional. Thus the arithmetic genus of X is the dimension of ${\displaystyle H^{1}(X,{\mathcal {O}}_{X})}$. By definition, the geometric genusofX is the dimension of H⁰(X, ω_X). Serre duality thus implies that the arithmetic genus and the geometric genus coincide. They will simply be called the genus of X.

Serre duality is also a key ingredient in the proof of the Riemann–Roch theorem. Since X is smooth, there is an isomorphism of groups

${\displaystyle {\begin{cases}\operatorname {Cl} ($X)\to \operatorname {Pic} (X)\\D\mapsto {\mathcal {O}}(D)\end{cases}}}