Surface Riemann-Roch and the Hodge Index Theorem — Examples

Given: a field $k$, the projective plane $X={\mathbb{P}}_k^2$ with its twisting sheaves ${\mathcal{O}}_X(d)$, a $k$-rational point $p\in X(k)$, the blowup $\pi :X^{\mathrm{\prime }}={\mathrm{Bl}}_{p}X\to X$ with exceptional curve $E={\pi }^{-1}(p)$, and the class $\mathrm{\ell }={\pi }^{\ast }{\mathcal{O}}_X(1)$.

The plane: $X$ is an integral regular projective surface over $k$ of pure dimension two. Its standard charts are the affine planes $\mathrm{Spec}k[y_1,y_2]$ with coordinate rings polynomial domains of dimension two and regular local rings, the charts are Noetherian and form a finite cover, and $X$ is projective over $k$ hence proper (Projective space is Proj of a polynomial ring, Relative projective space from standard charts, A polynomial ring in finitely many indeterminates over an integral domain is an integral domain, A polynomial ring in n variables over a field has dimension n, Dimension can be computed on an open cover, localisation and polynomial extension of regular rings, Integral schemes, Intersection numbers of Cartier divisors on a smooth projective surface, Projective morphisms before Proj, Projective morphisms are proper). As a smooth finite-type $k$-scheme, $X$ is locally factorial (Regular local rings are unique factorization domains, Locally factorial scheme).

Blowup calculus at a $k$-rational point: the residue degree is $r=[\kappa (p):k]=1$; $X^{\mathrm{\prime }}$ is an integral regular projective surface over $k$; $E$ is an effective Cartier divisor with $E\cong {\mathbb{P}}_k^1$ and ${\mathcal{O}}_E(E)\cong {\mathcal{O}}_{{\mathbb{P}}_k^1}(-1)$, so $E\cdot E=-r=-1$; and for all Cartier divisors $D,D^{\mathrm{\prime }}$ on $X$ one has $E\cdot {\pi }^{\ast }D=0$ and ${\pi }^{\ast }D\cdot {\pi }^{\ast }{D}^{\mathrm{\prime }}=D\cdot D^{\mathrm{\prime }}$ (The intersection matrix of a point blowup of a regular surface, Exceptional subscheme of a blowup, Intersection numbers of Cartier divisors on a smooth projective surface).

Cohomology of twists on the plane: $H^0(X,{\mathcal{O}}_X(m))$ is the degree-$m$ part of $k[x_0,x_1,x_2]$ for $m\ge 0$ and vanishes for $m<0$; $H^1(X,{\mathcal{O}}_X(m))=0$ for every $m$; $H^2(X,{\mathcal{O}}_X(m))=0$ for $m>-3$ (Global sections of projective twists, Intermediate cohomology of projective twists vanishes, Top cohomology of projective twists). Hence $\chi (X,{\mathcal{O}}_X)=1$, $\chi (X,{\mathcal{O}}_X(-1))=0$ and $\chi (X,{\mathcal{O}}_X(-2))=0$ (Euler characteristic of a coherent sheaf).

Lines are effective Cartier divisors with associated sheaf ${\mathcal{O}}_X(1)$: a nonzero linear form is a global section of ${\mathcal{O}}_X(1)$ (Global sections of projective twists), it is regular on the integral scheme $X$, and its zero scheme is an effective Cartier divisor $L$ with ${\mathcal{O}}_X(L)\cong {\mathcal{O}}_X(1)$ (A regular global section of an invertible sheaf glues to an effective Cartier divisor, Invertible sheaf of cartier divisor).

Total transform: for a reduced effective Cartier divisor $C$ on the regular surface $X$ with a closed point $p$ at which the multiplicity $m={\mathrm{mult}}_p(C)$ is finite and positive, ${\pi }^{\ast }C=C^{\mathrm{\prime }}+mE$ as effective Cartier divisors, where $C^{\mathrm{\prime }}$ is the strict transform (Total transform equals strict transform plus multiplicity times the exceptional divisor, Strict transform of a closed subscheme); passage to associated invertible sheaves uses ${\mathcal{O}}_{X^{\mathrm{\prime }}}({\pi }^{\ast }C)\cong {\pi }^{\ast }{\mathcal{O}}_X(C)$ (Pullback of a Cartier divisor computes the pullback of its line bundle, Pullback of a Cartier divisor).

Divisors and classes: a Weil divisor on a Noetherian normal scheme is a finite integral combination of prime divisors, the principal Weil divisor of $f\in K(X{)}^{\times }$ is ${\mathrm{div}}_W(f)={\sum }_Z{\mathrm{ord}}_Z(f)[Z]$, and $\mathrm{Cl}$ is the quotient by principal divisors (Weil divisor normal noetherian scheme, Principal weil divisor and class group, Order codimension one rational function). The order along a prime divisor with generic point $\xi$ is the valuation of the discrete valuation ring ${\mathcal{O}}_{X,\xi }$, which is a DVR because $X$ is normal and Noetherian (Height-one localizations of normal Noetherian domains are DVRs, Discrete valuations); orders are additive, vanish on units, and a uniformiser has order one.

On a locally factorial Noetherian integral scheme, Cartier and Weil divisors agree compatibly with principal divisors, and the canonical map $\mathrm{Pic}(X)\to \mathrm{Cl}(X)$ carrying $[{\mathcal{O}}_X(D)]$ to the class of the associated Weil divisor is an isomorphism (Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme); the map $D\mapsto [{\mathcal{O}}_X(D)]$ induces an isomorphism $\mathrm{CaDiv}(X)\mathrm{/}{\mathrm{Prin}}_C(X)\stackrel{\sim }{\to }\mathrm{Pic}(X)$ (On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group, Invertible sheaf of cartier divisor).

Off the centre: $\pi$ restricts to an isomorphism ${\pi }^{-1}(X\setminus \{p\})\to X\setminus \{p\}$ (The blowup is an isomorphism off the center), and ${\pi }^{-1}(X\setminus \{p\})=X^{\mathrm{\prime }}\setminus E$ (Exceptional subscheme of a blowup). For a reduced effective Cartier divisor $L$ through $p$ with strict transform $m$, this identifies $m\setminus E$ with $L\setminus \{p\}$ (Strict transform of a closed subscheme).

Standard charts: after a linear change of homogeneous coordinates taking the line $L$ to $V(x_0)$, $X\setminus L$ is the affine chart $U_0=\mathrm{Spec}k[x_1\mathrm{/}{x}_0,x_2\mathrm{/}{x}_0]\cong {\mathbb{A}}^2$, a polynomial domain over $k$ (Relative projective space from standard charts, Projective space is Proj of a polynomial ring, A polynomial ring in finitely many indeterminates over an integral domain is an integral domain).

In a polynomial ring $k[y_1,y_2]$ every height-one prime is principal, generated by an irreducible element, and every irreducible element is prime (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, Unique factorisation domain). Consequently every Weil divisor on $\mathrm{Spec}k[y_1,y_2]$ is principal: a prime divisor is $V(g)$ for an irreducible $g$, and for a finite combination ${\sum }_i{n}_i[V(g_i)]$ the rational function ${\prod }_i{g}_i^{n_i}$ has divisor ${\sum }_i{n}_i[V(g_i)]$, because each $g_i$ has order one along $V(g_i)$ and order zero along the other $V(g_j)$ by [F6], [F10].

Excision for class groups: if $Z$ is a prime divisor on an integral Noetherian normal scheme $X$ with $U=X\setminus Z$, restriction of Weil divisors (closure in $X$ of each prime divisor of $U$) is surjective with kernel $\mathbb{Z}[Z]$, and principal divisors restrict to principal divisors; hence the induced map $\mathrm{Cl}(X)\to \mathrm{Cl}(U)$ is surjective and its kernel is the image of $\mathbb{Z}[Z]$ (Weil divisor normal noetherian scheme, Principal weil divisor and class group, Order codimension one rational function). The same statement holds with $Z$ replaced by a union of finitely many prime divisors, with kernel the direct sum of their classes. To verify the class-group kernel, a prime divisor meeting $U$ has the same codimension-one local ring at its generic point on $U$ and on the whole scheme, so restriction preserves its valuation and the divisor of every rational function. Prime divisors of $U$ extend by closure, proving surjectivity. If a divisor restricts to ${\mathrm{div}}_U(f)$, use the same $f$ in the common function field and subtract its divisor on the whole scheme; the difference is supported on the removed prime divisors. Conversely those boundary divisors restrict to zero. This proves the asserted exactness.

Intersection numbers depend only on the linear equivalence classes (Intersection numbers of Cartier divisors on a smooth projective surface, The surface intersection product is symmetric and bilinear, Cartier divisor); in particular a Cartier divisor linearly equivalent to $0$ has zero intersection with every Cartier divisor.

A nonzero global section of an invertible sheaf on the integral scheme $X$ is regular with effective Cartier zero scheme (A regular global section of an invertible sheaf glues to an effective Cartier divisor); duals and tensor products of invertible sheaves are invertible and ${\mathcal{L}}^{\vee }\otimes \mathcal{L}\cong {\mathcal{O}}_X$ (Dual of a line bundle is its tensor inverse, Invertible sheaves).

The Axiom of Choice is inherited from the projective-space, blowup and divisor-class suppliers above; all selections below are finite (a line, a point, finitely many prime divisors).

Given: a field $k$ and the product $X={\mathbb{P}}_k^1{\times }_{\mathrm{Spec}k}{\mathbb{P}}_k^1$ with its projections.

Chart data: the projective line has the two standard charts $U_0=\mathrm{Spec}k[t]$ and $U_1=\mathrm{Spec}k[u]$, glued along $D(t)=D(u)$ by $t=u^{-1}$ (Two-affine projective line and its twists, Relative projective space from standard charts). In particular ${\mathbb{P}}_k^1$ is covered by two affine schemes and the structure morphism ${\mathbb{P}}_k^1\to \mathrm{Spec}k$ is quasi-compact.

Products of affine schemes and polynomial rings: $\mathrm{Spec}B{\times }_{\mathrm{Spec}A}\mathrm{Spec}C\cong \mathrm{Spec}(B{\otimes }_{A}C)$ for ring maps $A\to B$, $A\to C$ (Affine fibre products are spectra of tensor products); $k[t]{\otimes }_{k}k[u]\cong k[t,u]$ as iterated polynomial rings (Polynomial rings in finitely many commuting indeterminates by iteration, A polynomial ring on a finite ordered family agrees canonically with the iterated polynomial-ring construction); $k[t,u]$ is a domain (A polynomial ring in finitely many indeterminates over an integral domain is an integral domain), Noetherian (If $R$ is Noetherian then $R[x_1,\dots ,x_n]$ is Noetherian for every $n\in \mathbb{N}$) and of Krull dimension two (A polynomial ring in n variables over a field has dimension n, Krull dimension of a nonzero ring).

Gluing fibre products: products of the open pieces glue to the fibre product, with no separatedness hypothesis (Gluing fibre products along open covers).

Irreducibility: a nonempty topological space is irreducible exactly when every two of its nonempty open subsets meet (Irreducibility via nonempty open subsets, connectedness and open subspaces). For a domain $A$ the spectrum $\mathrm{Spec}A$ is irreducible with generic point the zero ideal, and irreducible closed subsets correspond to prime ideals (A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point); a scheme is integral when it is nonempty, reduced and irreducible (Integral schemes), and reducedness means the nilpotent ideal sheaf vanishes, equivalently all local rings are reduced (The reduction of a scheme).

Dimension: for a Noetherian space covered by finitely many open subspaces, the chain dimension is the supremum of the dimensions of the pieces (Dimension can be computed on an open cover, Chain dimension and the empty-space convention).

Smoothness: ${\mathbb{P}}_k^1$ is a smooth curve over $k$ (Projective-line curve and divisor basics), ${\mathbb{P}}_k^1\to \mathrm{Spec}k$ is of finite type (Projective space is of finite type over its base), and for any field $k$ and finite-type $k$-schemes $X,Y$ smooth over $k$, the product $X{\times }_{k}Y$ is smooth over $k$ in the local-standard-smooth convention; a product of finite-type $k$-schemes is of finite type over $k$ (Products preserve smoothness, Finite type under base change and products over a field, Smoothness over a field by geometric regularity).

Projectivity: for $m=n=1$ and $S=\mathrm{Spec}k$ the Segre construction gives a closed immersion $\sigma :X\hookrightarrow {\mathbb{P}}_k^3$ with ${\sigma }^{\ast }\mathcal{O}(1)\cong {\mathrm{p}\mathrm{r}}_1^{\ast }\mathcal{O}(1)\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }\mathcal{O}(1)$ (Segre embedding and its line bundle); composing with the projection exhibits the structure morphism $X\to \mathrm{Spec}k$ as projective in the H-projective convention, hence proper (Projective morphisms before Proj, Projective morphisms are proper).

Projections of the product: ${\mathbb{P}}_k^1\to \mathrm{Spec}k$ is flat, because its two standard charts are standard smooth $k$-algebras and standard smooth algebras are flat (Standard smooth algebras are finitely presented and flat, Flat morphism of schemes); it is proper (Finite-dimensional projective space is proper over every base) and locally of finite presentation, its charts being finitely presented $k$-algebras (Standard smooth algebras are finitely presented and flat, Locally finite presentation morphisms). Each projection ${\mathrm{p}\mathrm{r}}_i$ is the base change of this morphism along the structure morphism of the other factor, hence flat (Flatness is stable under arbitrary base change), proper (Properness survives arbitrary base change) and locally of finite presentation (Local finiteness conditions under base change); it is quasi-compact because the preimage of each standard chart is the union of the two affine charts lying over it (Quasi-compact and quasi-separated morphisms). Hence each projection is of finite presentation.

The Axiom of Choice is inherited from the scheme, product and Segre suppliers above; only the two-element chart cover and the four chart products are used below.

Given: a field $k$, the surface $X={\mathbb{P}}_k^1{\times }_k{\mathbb{P}}_k^1$ with its projections, the ruling fibres $\mathrm{\ell }={\mathrm{p}\mathrm{r}}_1^{-1}(\mathrm{\infty })$ and $m={\mathrm{p}\mathrm{r}}_2^{-1}(\mathrm{\infty })$, and the diagonal $\mathrm{\Delta }\subseteq X$.

By the structure lemma, $X$ is an integral smooth projective surface over $k$, so it is Noetherian, regular and (being smooth of finite type over $k$) locally factorial; the projections ${\mathrm{p}\mathrm{r}}_1,{\mathrm{p}\mathrm{r}}_2$ are flat and proper, and $\pi :{\mathbb{P}}_k^1\to \mathrm{Spec}k$ is flat (The product of two projective lines is an integral smooth projective surface, Regular local rings are unique factorization domains, Locally factorial scheme, Flat morphism of schemes). The intersection product on $X$ is defined, symmetric and $\mathbb{Z}$-bilinear (Intersection numbers of Cartier divisors on a smooth projective surface, The surface intersection product is symmetric and bilinear).

Pullbacks of Cartier divisors along flat morphisms are defined: a regular section pulls back to a regular section under a flat morphism, so the pullback datum of Pullback of a Cartier divisor exists; and for a Cartier divisor $D$ one has ${\mathcal{O}}_X(f^{\ast }D)\cong f^{\ast }{\mathcal{O}}_Y(D)$ (Pullback of a Cartier divisor, Pullback of a Cartier divisor computes the pullback of its line bundle, Flat morphism of schemes). The point $\mathrm{\infty }\in {\mathbb{P}}_k^1$ is an effective Cartier divisor with ${\mathcal{O}}_{{\mathbb{P}}^1}(\mathrm{\infty })\cong {\mathcal{O}}_{{\mathbb{P}}^1}(1)$: the coordinate $x_0$ is a nonzero global section of $\mathcal{O}(1)$ (Global sections of projective twists) with zero scheme $\mathrm{\infty }$ (A regular global section of an invertible sheaf glues to an effective Cartier divisor, Relative projective space from standard charts).

Intersection with a curve is the degree of the restriction: for an effective Cartier divisor $C$ and any Cartier divisor $D$, $C\cdot D={\mathrm{deg}}_C({\mathcal{O}}_X(D){\mathrm{\mid }}_C)$, where ${\mathrm{deg}}_C(\mathcal{N})=\chi (C,\mathcal{N})-\chi (C,{\mathcal{O}}_C)$ (Intersection with a curve is the degree of the restriction, Degree of an invertible sheaf on a proper one-dimensional scheme, Euler characteristic of a coherent sheaf). On ${\mathbb{P}}_k^1$ one has $h^0(\mathcal{O})=1$, $h^0(\mathcal{O}(1))=2$ and $h^1(\mathcal{O})=h^1(\mathcal{O}(1))=0$ (Global sections of projective twists, Top cohomology of projective twists), so ${\mathrm{deg}}_{{\mathbb{P}}^1}({\mathcal{O}}_{{\mathbb{P}}^1}(1))=1$.

Class groups: $X$ is a locally factorial Noetherian integral scheme, so $\mathrm{Pic}(X)\cong \mathrm{Cl}(X)$ compatibly with the Weil divisor of a Cartier divisor, and $\mathrm{CaDiv}(X)\mathrm{/}{\mathrm{Prin}}_C(X)\cong \mathrm{Pic}(X)$ (Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme, On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group, Weil divisor normal noetherian scheme, Principal weil divisor and class group).

Excision: for $U=X\setminus (\mathrm{\ell }\cup m)$ the restriction of Weil divisors (closure in $X$ of each prime divisor of $U$) is surjective with kernel $\mathbb{Z}[\mathrm{\ell }]\oplus \mathbb{Z}[m]$, and principal divisors restrict to principal divisors; hence $\mathrm{Cl}(X)\to \mathrm{Cl}(U)$ is surjective with kernel generated by $[\mathrm{\ell }],[m]$ (Weil divisor normal noetherian scheme, Principal weil divisor and class group, Order codimension one rational function, Height-one localizations of normal Noetherian domains are DVRs). Moreover $U={\mathrm{p}\mathrm{r}}_1^{-1}(U_0)\cap {\mathrm{p}\mathrm{r}}_2^{-1}(V_0)=U_0{\times }_k{V}_0$ for the standard affine charts of the two factors, and $U_0{\times }_k{V}_0\cong \mathrm{Spec}k[y_1]{\times }_k\mathrm{Spec}k[y_2]\cong \mathrm{Spec}k[y_1,y_2]$ (Relative projective space from standard charts, Affine fibre products are spectra of tensor products, Projective space is Proj of a polynomial ring). To verify the class-group kernel, a prime divisor meeting $U$ has the same codimension-one local ring at its generic point on $U$ and on the whole scheme, so restriction preserves its valuation and the divisor of every rational function. Prime divisors of $U$ extend by closure, proving surjectivity. If a divisor restricts to ${\mathrm{div}}_U(f)$, use the same $f$ in the common function field and subtract its divisor on the whole scheme; the difference is supported on the removed prime divisors. Conversely those boundary divisors restrict to zero. This proves the asserted exactness.

In $k[y_1,y_2]$ every height-one prime is principal and every irreducible element is prime, and every Weil divisor is a finite combination of prime divisors, so every Weil divisor on $U\cong \mathrm{Spec}k[y_1,y_2]$ is principal: for a combination ${\sum }_i{n}_i[V(g_i)]$ the rational function ${\prod }_i{g}_i^{n_i}$ has divisor ${\sum }_i{n}_i[V(g_i)]$, since each $g_i$ has order one along $V(g_i)$ and order zero along the other primes (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, Unique factorisation domain, Order codimension one rational function, Discrete valuations). Hence $\mathrm{Cl}(U)=0$.

Diagonal: the pullbacks ${\mathrm{p}\mathrm{r}}_1^{\ast }{x}_j$ and ${\mathrm{p}\mathrm{r}}_2^{\ast }{y}_j$ of the coordinate sections are global sections of ${\mathrm{p}\mathrm{r}}_1^{\ast }\mathcal{O}(1)$ and ${\mathrm{p}\mathrm{r}}_2^{\ast }\mathcal{O}(1)$ whose zero schemes are the fibres ${\mathrm{p}\mathrm{r}}_1^{-1}(V(x_j))$ and ${\mathrm{p}\mathrm{r}}_2^{-1}(V(y_j))$; the section $s={\mathrm{p}\mathrm{r}}_1^{\ast }{x}_0\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }{y}_1-{\mathrm{p}\mathrm{r}}_1^{\ast }{x}_1\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }{y}_0$ of the invertible sheaf ${\mathrm{p}\mathrm{r}}_1^{\ast }\mathcal{O}(1)\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }\mathcal{O}(1)$ is nonzero, and on the two equal-index charts its coefficient is the difference of the affine coordinates, while on a mixed chart it is, up to sign, $1-tu$. The equal-index equations identify the coordinates; on a mixed chart $tu=1$ identifies the two projective points on their overlap. Thus its zero scheme is the diagonal $\mathrm{\Delta }$ (Zero scheme of a line-bundle section, A regular global section of an invertible sheaf glues to an effective Cartier divisor, Pullback of a Cartier divisor computes the pullback of its line bundle, Relative projective space from standard charts). Consequently $\mathrm{\Delta }$ is an effective Cartier divisor with ${\mathcal{O}}_X(\mathrm{\Delta })\cong {\mathrm{p}\mathrm{r}}_1^{\ast }\mathcal{O}(1)\otimes {\mathrm{p}\mathrm{r}}_2^{\ast }\mathcal{O}(1)$.

The Axiom of Choice is inherited from the intersection, pullback and class-group suppliers above; all computations use the two ruling fibres, the diagonal and finitely many chart functions.

Given: a field $k$, an integral smooth projective surface $X$ over $k$, an ample invertible sheaf $H$ on $X$, and the real Neron-Severi space ${\mathrm{N}}_{\mathbb{R}}^1(X)$ with its intersection form.

Ample classes on a projective surface exist: the H-projective structure of $X$ gives a closed immersion $i:X\hookrightarrow {\mathbb{P}}_k^n$ with ${\mathcal{O}}_X(1)=i^{\ast }{\mathcal{O}}_{{\mathbb{P}}^n}(1)$ closed H-very ample relative to $\mathrm{Spec}k$ (Projective morphisms before Proj, Relative very ampleness in the finite projective-space convention); such a class is H-very ample and ample (Relative very ampleness implies relative ampleness, Absolute ampleness by affine section opens).

Positivity: for every ample $H$ one has $H\cdot H>0$ (Ample divisors meet nonzero effective divisors positively, case (2)); the intersection product is symmetric and $\mathbb{Z}$-bilinear (The surface intersection product is symmetric and bilinear).

Numerical equivalence: a class $[M]\in {\mathrm{N}}_{\mathbb{R}}^1(X)$ is zero exactly when $M$ is numerically trivial, i.e. $M\cdot N=0$ for every invertible $\mathcal{N}$; thus $[H]\mathrm{\ne }0$ whenever $H\cdot H>0$ (Numerical equivalence and the Neron-Severi space of a surface, Invertible sheaves).

A real form is negative semidefinite when $q(v)\le 0$ for all $v$, and negative definite when $q(v)<0$ for all $v\mathrm{\ne }0$; a single nonzero class with positive square refutes both properties (Numerical equivalence and the Neron-Severi space of a surface for the self-intersection form).

The Axiom of Choice is inherited from the ample-positivity and embedding suppliers of [F1]–[F2]; the sheaf $H$ is a single given object.

Given: a field $k$, the plane $X={\mathbb{P}}_k^2$, a $k$-rational point $p$, the blowup $X^{\mathrm{\prime }}={\mathrm{Bl}}_{p}X$ with exceptional curve $E$, the class $\mathrm{\ell }={\pi }^{\ast }{\mathcal{O}}_X(1)$, and the strict transform $m$ of a line through $p$.

Blowup data: $X^{\mathrm{\prime }}$ is an integral smooth projective surface over $k$; $\mathrm{Pic}(X^{\mathrm{\prime }})=\mathbb{Z}\mathrm{\ell }\oplus \mathbb{Z}E$; ${\mathrm{\ell }}^2=1$, $\mathrm{\ell }\cdot E=0$, $E^2=-1$; and the strict transform $m$ of a line through $p$ satisfies ${\pi }^{\ast }L=m+E$ for such a line $L$, so $m=\mathrm{\ell }-E$ in the Picard group, $m\cdot E=1$ and $m^2=0$ by bilinearity (The Picard group of a point blowup of the projective plane, The intersection matrix of a point blowup of a regular surface, Total transform equals strict transform plus multiplicity times the exceptional divisor, The surface intersection product is symmetric and bilinear, Effective cartier divisor).

Numerical space and definiteness data: ${\mathrm{N}}_{\mathbb{R}}^1(X^{\mathrm{\prime }})$ is the real extension of $\mathrm{Pic}(X^{\mathrm{\prime }})$ modulo numerical equivalence; the parametrization $(a,b)\mapsto a\mathrm{\ell }+bE$ is an isomorphism at the level of Picard groups, and the intersection form on the real space is the bilinear extension, with inertia, rank and signature as in Positive and negative definiteness, the inertia $(p,q,r)$, rank $p+q$, and signature $p-q$ of a real symmetric bilinear or quadratic form (Numerical equivalence and the Neron-Severi space of a surface, Intersection numbers of Cartier divisors on a smooth projective surface).

Hodge index theorem and its corollary: if $H\cdot H>0$ and $L\cdot H=0$ for invertible sheaves, then $L\cdot L\le 0$ with equality exactly for numerically trivial $L$; and for $H\cdot H>0$ the form is negative definite on the primitive part $h^{\perp }$ of the real Neron-Severi space (The Hodge index theorem for smooth projective surfaces, Negative definiteness of the primitive part of the Neron-Severi space).