By the factorization theorem (Zariski's main theorem, and the factorization of birational maps of surfaces), every birational map between smooth projective surfaces is a composite of blowups at points and their inverses. A quantity unchanged by one blowup at a point is therefore a birational invariant of smooth projective surfaces.
What a blowup does to $\Pic$, to $K$, and to the intersection form (Tag FE-SRFBLOW)
Let \(\pi \colon \tilde{X} = \Bl_p X \to X\) be the blowup of a smooth surface at a point, with exceptional curve \(E \cong \PP^1\). Then \[ \Pic \tilde{X} = \pi^* \Pic X \oplus \ZZ E , \] and the intersection numbers are \[ \pi^*C \cdot \pi^*D = C \cdot D , \qquad \pi^*C \cdot E = 0 , \qquad E^2 = -1 . \] For the canonical divisor, \[ K_{\tilde{X}} = \pi^* K_X + E , \qquad K_{\tilde{X}}^2 = K_X^2 - 1 , \] and \(\chi(\OO)\), \(p_g\) and \(q\) are unchanged.
(Hironaka resolution (characteristic zero))
Let \(k = \CC\).
For every projective variety \(Y\) there exist a smooth projective variety \(Y'\) and a birational morphism \(\varphi \colon Y' \to Y\) such that \(\varphi\) is an isomorphism over \(\reg Y\).
For every smooth projective variety \(X\) and projective subvariety \(Y \subset X\) there exist a smooth projective variety \(W\) and a birational morphism \(\varphi \colon W \to X\) such that \(\varphi^{-1}(Y)\) is a divisor with simple normal crossings in \(W\).
If \(Y \subset X\) as in (b) is a hypersurface, there is a unique irreducible component \(Y'\) of \(\varphi^{-1}(Y)\) dominating \(Y\), and \(\varphi\vert_{Y'} \colon Y' \to Y\) is a desingularization as in (a); it is called an embedded desingularization of \(Y\).
In (a) and (b) the morphism \(\varphi\) can be chosen as a composition of blowups with smooth centres.
In particular every projective variety admits a desingularization by a sequence of blowups with smooth centres, and every subvariety can be made SNC after such a sequence.
(Strict transform)
For \(C \subseteq X\) passing through \(p\) with multiplicity \(m\), the strict transform \(\tilde{C}\) is the closure of \(\pi^{-1}(C \setminus p)\), and \[ \pi^* C = \tilde{C} + mE , \qquad \tilde{C}^2 = C^2 - m^2 , \qquad p_a(\tilde{C}) = p_a(C) - \binom{m}{2} . \]
\(E^2 = -1\) is the fact the rest hangs on, and it follows from \(\pi^*C \cdot E = 0\) applied to a curve through \(p\): writing \(\pi^*C = \tilde{C} + E\) for a smooth \(C\) gives \(0 = \tilde{C}\cdot E + E^2 = 1 + E^2\).
The genus drop is the tool for resolving plane curve singularities: blowing up an ordinary \(m\)-fold point removes \(\binom{m}{2}\) from the arithmetic genus, which matches the delta invariant, so repeated blowups compute the geometric genus. A node has \(m=2\) and drops the genus by one.
\(K^2\) decreasing by one while \(\chi(\OO)\) stays fixed is the numerical fingerprint of a blowup, and it is what Noether’s formula balances: \(c_2\) goes up by one.
Under a blowup at a point, \(K^2\) drops by one and \(\rho\) rises by one, while \(\chi(\OO)\), \(p_g\), and \(q\) are unchanged.
For a curve \(C\) with an ordinary \(m\)-fold point at the centre of the blowup, the strict transform has \(p_a(\tilde C)=p_a(C)-\binom m2\), and \(\binom m2\) is the \(\delta\)-invariant of an ordinary \(m\)-fold point.
Every curve has a unique smooth projective model (Tag T-CRVMINMOD)
Let \(k\) be an algebraically closed field. The following categories are equivalent:
smooth projective curves over \(k\) with dominant morphisms;
quasiprojective curves over \(k\) with dominant rational maps;
finitely generated field extensions \(K/k\) of transcendence degree \(1\), with \(k\)-algebra homomorphisms, with arrows reversed.
In particular every curve is birational to a smooth projective curve, unique up to isomorphism, and two smooth projective curves are birational exactly when they are isomorphic.
The smooth projective model of the function field \(K\) is built from its discrete valuation rings: its closed points are the DVRs of \(K\) containing \(k\), and the normalization of any projective model of \(K\) realizes it. A rational map from a smooth curve to a projective variety extends to a morphism, by the valuative criterion applied at each point, which is why birational smooth projective curves are isomorphic. In dimension \(2\) this uniqueness fails: blowing up a point gives a birational smooth projective surface that is not isomorphic, and minimal models are reached by contracting \((-1)\)-curves.
Castelnuovo's contractibility criterion (Tag T-SRFCAST)
Let \(X\) be a smooth surface, separated and of finite type over an algebraically closed field \(k\), and let \(E \subseteq X\) be an integral curve that is proper over \(k\). There exist a smooth surface \(Y\) over \(k\), a point \(p \in Y\) and a morphism \(\sigma \colon X \to Y\) that is the blowup of \(Y\) at \(p\) with exceptional curve \(E\) if and only if \(E \cong \PP^1\) and \(E^2 = -1\). Such a curve is a \((-1)\)-curve, or exceptional curve of the first kind. If \(X\) is projective, then \(Y\) is projective [@Har10a, Theorem V.5.7].
This converts a numerical condition into a birational morphism, and it is what makes the minimal model program work in dimension two: contract \((-1)\)-curves until none remain, and the result is a minimal surface. Since each contraction raises \(K^2\) by one and \(\rho\) drops by one, the process terminates.
By adjunction, \(E \cong \PP^1\) with \(E^2 = -1\) is equivalent to \(E^2 = -1\) and \(K \cdot E = -1\), which is the form the criterion is usually applied in.
A surface can have infinitely many \((-1)\)-curves: the blowup of \(\PP^2\) at \(9\) points in general position, in the sense that no three of the points are collinear after any finite sequence of quadratic transformations centred at three of them, has infinitely many (P-AGH5415GENPOSITION, part (e)), while for \(r = 7\) and \(r = 8\) points in general position there are exactly \(56\) and \(240\) (part (d)). The minimal model is then not unique in general — for rational surfaces one can reach both \(\PP^2\) and the Hirzebruch surfaces — which is exactly the dimension-two failure that makes the classification of rational and ruled surfaces a separate theorem.
Contracting \((-1)\)-curves on a smooth projective surface one at a time lowers \(\rho\) by one at each step, so after finitely many steps the result is a minimal surface.
A surface need not have finitely many \((-1)\)-curves: the blowup of \(\PP^2\) at \(n\ge9\) points in general position has infinitely many.
Zariski's main theorem, and the factorization of birational maps of surfaces (Tag T-SRFZMT)
Let \(f \colon X \to Y\) be a birational projective morphism of Noetherian integral schemes with \(Y\) normal. Then every fibre \(f^{-1}(y)\) is connected.
(Elimination of indeterminacy (Hironaka))
Let \(X\) and \(Y\) be smooth projective varieties over an algebraically closed field \(k\), and \(F \colon X \dashrightarrow Y\) a birational map. Assume that \(k\) has characteristic \(0\), or that \(\dim X = 2\). Then there exist a smooth projective variety \(W\) and birational morphisms \(g \colon W \to X\) and \(f \colon W \to Y\) fitting into
with \(F \circ g = f\) on the open subset of \(W\) where \(F \circ g\) is defined.
(Factorization)
Let \(f \colon X' \to X\) be a birational morphism of smooth projective surfaces and let \(p\) be a fundamental point of \(f^{-1}\). Then \(f\) factors through the blowup of \(X\) at \(p\). Consequently every birational map of smooth projective surfaces is a finite sequence of blowups followed by a finite sequence of blowdowns.
The connectedness theorem supplies the fibre structure used in the surface factorization statement. A birational morphism of smooth surfaces is an isomorphism away from finitely many points, and at each such point the fibre is connected of dimension one — so it is a curve that gets contracted, and contraction of a curve to a smooth point is a blowup.
Fundamental points are the points where the inverse map fails to be defined: the map is a morphism on a largest open \(U\), and \(X \setminus U\) is that finite set. The total transform of such a point is what Zariski’s theorem says is connected and positive-dimensional.
The factorization statement is the reason surfaces are classified up to birational equivalence rather than isomorphism, and why the invariants that matter — \(p_g\), \(q\), the plurigenera, \(\kappa\) — are exactly those unchanged by blowup.
In the factorization theorem, Zariski’s connectedness theorem makes each positive-dimensional fibre of a birational morphism of smooth projective surfaces a connected curve, which is contracted to a point.
The surfaces themselves
Rational and ruled surfaces, and the Hirzebruch surfaces (Tag D-SRFRULED)
A surface \(X\) is ruled over a curve \(C\) if it is birational to \(C \times \PP^1\); it is geometrically ruled if there is a morphism \(X \to C\) whose every fibre is \(\PP^1\), equivalently \(X = \PP(\mathcal{E})\) for a rank-two bundle \(\mathcal{E}\) on \(C\). \(X\) is rational if it is birational to \(\PP^2\).
(Hirzebruch surfaces)
\(\FF_n = \PP(\OO_{\PP^1} \oplus \OO_{\PP^1}(n))\) is the geometrically ruled surface over \(\PP^1\) with a section \(C_0\) satisfying \(C_0^2 = -n\). \(\FF_0 = \PP^1 \times \PP^1\) and \(\FF_1 = \Bl_p \PP^2\). The minimal rational surfaces are exactly \(\PP^2\) and \(\FF_n\) for \(n \neq 1\).
(Castelnuovo's rationality criterion)
A smooth projective surface over \(\CC\) is rational if and only if \(q = 0\) and \(P_2 = h^0(2K) = 0\).
The section \(C_0\) of \(\FF_1\) is a \((-1)\)-curve, and contracting it gives \(\PP^2\), so \(\FF_1\) is not minimal. The minimal rational surfaces \(\PP^2\), \(\FF_0\), \(\FF_2\), \(\FF_3,\ldots\) are pairwise non-isomorphic and birational to each other, so a rational surface has no unique minimal model.
The conditions \(p_g=q=0\) do not imply rationality: Enriques surfaces satisfy them but have \(P_2=1\), while Castelnuovo’s criterion requires \(q=P_2=0\).
A smooth projective surface over \(\CC\) has Kodaira dimension \(\kappa = -\infty\), that is, all plurigenera \(P_m=h^0(mK)\) with \(m\ge1\) vanish, if and only if it is ruled; rational surfaces are the surfaces ruled over \(\PP^1\).
Rational and ruled surfaces are the \(\kappa = -\infty\) case. Enriques surfaces have \(p_g=q=0\) and \(P_2=1\), so by Castelnuovo’s criterion they are not rational.
The cubic surface and its $27$ lines (Tag FE-SRFCUBIC)
A smooth cubic surface \(S \subseteq \PP^3\) is the blowup of \(\PP^2\) at \(6\) points in general position, embedded by the anticanonical system \(\abs{-K_S} = \abs{3H - \sum_{i=1}^{6} E_i}\). So \(\Pic S = \ZZ^7\) with basis \(H, E_1, \dots, E_6\), and \(K_S^2 = 9 - 6 = 3 = \deg S\).
(The 27 lines)
A line on \(S\) is a class \(L\) with \(L^2 = -1\) and \(L \cdot K_S = -1\), and there are exactly \(27\):
the \(6\) exceptional curves \(E_i\);
the \(15\) strict transforms \(H - E_i - E_j\) of the lines through two of the points;
the \(6\) strict transforms \(2H - \sum_{j \neq i} E_j\) of the conics through five of the points.
The count \(6 + 15 + 6 = 27\) is the answer to give, and it is a computation in \(\Pic\), not a geometric miracle: adjunction says a line has \(2g-2 = -2 = L^2 + L\cdot K\), so \(L^2 = -1\) and \(L \cdot K = -1\), and enumerating integer solutions in the basis gives exactly those three families.
A line on \(S\) is a \((-1)\)-curve, so each of the \(27\) can be contracted; different choices of six pairwise disjoint lines realise \(S\) as a blowup of \(\PP^2\) in different ways, which is why the configuration has a large symmetry group, the Weyl group of \(E_6\) of order \(51840\).
That \(S\) is anticanonically embedded is what makes it a del Pezzo surface of degree \(3\); the same analysis with \(9 - d\) blown-up points gives the del Pezzo surfaces of each degree, and the line counts \(27\), \(16\), \(10\) follow the same way.
A smooth cubic surface is the blowup of \(\PP^2\) at six points in general position, embedded by \(\abs{-K}\). Its lines are the classes \(L\in\Pic\) with \(L^2=L\cdot K=-1\), and there are \(27\) of them. Each line is a \((-1)\)-curve; each of the \(72\) sets of six pairwise disjoint lines contracts to give a morphism onto \(\PP^2\).
Kodaira dimension and the Enriques classification of surfaces (Tag T-SRFKOD)
The Kodaira dimension of a smooth projective variety \(X\) is \[ \kappa(X) = \operatorname{trdeg}_k \bigoplus_{n \geq 0} H^0(X, \OO_X(nK_X)) - 1 , \] set to \(-\infty\) when every plurigenus vanishes. It is a birational invariant, so for surfaces it is computed on a minimal model.
(Zariski desingularization of surfaces)
Every projective surface \(Y\) admits a desingularization by a finite sequence that alternates normalizations and blowups of maximal ideals. The resulting smooth surface is projective.
(Minimal resolution of normal surface singularities)
Let \(Y\) be a normal projective surface. There exists a unique minimal resolution \(\varphi \colon X \to Y\) such that \(X\) is smooth projective, \(\varphi\) is an isomorphism over \(\reg Y\), and every other desingularization \(\varphi' \colon X' \to Y\) with the same two properties factors as \(\varphi' = \varphi \circ \sigma\) where \(\sigma \colon X' \to X\) is a composition of blowups of points over the exceptional locus of \(\varphi\).
(Enriques classification)
A minimal smooth projective surface \(X\) over \(\CC\) falls into exactly one of:
\(\kappa(X) = -\infty\) iff \(|12K_X| = \varnothing\) iff \(X\) is \(\PP^2\) or a minimal ruled surface;
\(\kappa(X) = 0\) iff \(|12K_X| = \{0\}\) iff \(X\) is K3, Enriques, abelian, or bielliptic;
\(\kappa(X) = 1\) iff \(X\) is a properly elliptic surface;
\(\kappa(X) = 2\) iff \(X\) is of general type.
In particular \(\kappa = -\infty\) are the rational and ruled surfaces, \(\kappa = 0\) are the four classes above, \(\kappa = 1\) are elliptic, and \(\kappa = 2\) are general type.
The organising idea is that \(\kappa\) measures how much of the surface the pluricanonical systems can see: nothing at \(-\infty\), a point at \(0\), a curve at \(1\), the whole surface at \(2\). For curves the same invariant gives the trichotomy \(g=0\), \(g=1\), \(g \geq 2\), and the surface classification is its two-dimensional analogue.
The \(\kappa = 0\) row is the one to be able to separate, by \((p_g, q)\): K3 has \((1,0)\) with \(K \sim 0\) and simply connected; Enriques has \((0,0)\) with \(2K \sim 0\), and is a free quotient of a K3 by an involution; abelian has \((1,2)\); bielliptic has \((0,1)\) and is a quotient of a product of elliptic curves.
Minimality is what makes the statement clean, and blowing up is exactly the operation that leaves \(\kappa\) alone while changing \(K^2\) — which is why the classification is stated for minimal models and read off as a birational statement.
Over an algebraically closed field of characteristic \(p > 0\) the classification of Bombieri and Mumford has the same four values of \(\kappa\), but the \(\kappa = 0\) and \(\kappa = 1\) classes change: in characteristic \(2\) and \(3\) there are quasi-hyperelliptic surfaces and quasi-elliptic fibrations, and in characteristic \(2\) there are non-classical Enriques surfaces with \(h^1(\OO_X) = 1\). So the list above is a theorem for complex surfaces, equivalently for surfaces over an algebraically closed field of characteristic \(0\).
A curve of genus \(g\) has \(\kappa=-\infty\), \(0\), or \(1\) according as \(g=0\), \(g=1\), or \(g\ge2\). For minimal surfaces with \(\kappa=0\), the pair \((p_g,q)\) is \((1,0)\) for K3, \((0,0)\) for Enriques, \((1,2)\) for abelian, and \((0,1)\) for bielliptic surfaces.
Birational invariants
Plurigenera and irregularity are birational invariants (Tag T-BIRINV)
Let \(X\) be a smooth projective variety of dimension \(n\) over an algebraically closed field.
The geometric genus is \(p_g(X) = h^0(X, \omega_X)\), and the \(m\)-th plurigenus is \(P_m(X) = h^0(X, \omega_X^{\otimes m})\) for \(m \geq 1\).
The irregularity is \(q(X)=h^1(X,\OO_X)\); for surfaces this is also \(p_g(X)-p_a(X)\) [@Har10a, Remark III.7.12.3]. The number of regular one-forms is \(h^{1,0}(X)=h^0(X,\Omega^1_{X/k})\). In characteristic zero, Hodge symmetry gives \(q(X)=h^{1,0}(X)\) [@Har10a, Appendix B].
\(X\) is of general type if \(P_m(X)\) grows like a positive multiple of \(m^n\), that is, \(X\) has Kodaira dimension \(n\).
Let \(X\) and \(X'\) be birational smooth projective varieties of dimension \(n\) over the same algebraically closed field \(k\). For every \(0\le p\le n\) and \(m\ge1\), birational pullback gives an isomorphism \[ H^0(X,(\Omega^p_{X/k})^{\otimes m}) \cong H^0(X',(\Omega^p_{X'/k})^{\otimes m}), \qquad\Omega^p_{X/k}=\bigwedge^p\Omega_{X/k}. \] In particular \(p_g\), every \(P_m\), and every \(h^{p,0}\) are birational invariants in any characteristic. The inverse pullbacks are constructed in P-AGH288PLURIGENUS, following the method of [@Har10a, Theorem II.8.19 and Exercise II.8.8].
(Characteristic-zero Hodge comparison)
For smooth projective varieties in characteristic zero, Hodge symmetry gives \[ h^i(X,\OO_X)=h^0(X,\Omega^i_{X/k}) \] [@Har10a, Appendix B]. For a general characteristic-zero ground field, descend the smooth projective variety to a subfield finitely generated over \(\QQ\) and embed that subfield in \(\CC\); the complex comparison gives the equality, and coherent cohomology commutes with both field extensions [@Har10a, Proposition III.9.3]. Consequently the preceding theorem also gives birational invariance of \(q(X)=h^1(X,\OO_X)\), of \(\chi(X,\OO_X)=\sum_i(-1)^i h^i(X,\OO_X)\), and of the arithmetic genus \[ p_a(X)=(-1)^n\bigl(\chi(X,\OO_X)-1\bigr) \] [@Har10a, Chapter I, §7 and Exercise III.5.2]. The Euler characteristic and arithmetic genus are distinct quantities.
For a smooth projective surface, blowing up a point gives \(K_{\widetilde X}=\pi^*K_X+E\), by P-AGH285BLOWUPCANON. Since \(E^2=-1\) and \((\pi^*K_X).E=0\), one obtains \(K_{\widetilde X}^2=K_X^2-1\) [@Har10a, Chapter V, §3]. Thus the self-intersection of the canonical divisor is not a birational invariant.
A K3 surface
The Fermat quartic is a K3 surface containing $48$ lines (Tag FE-SRFFERMATQUARTIC)
The Fermat quartic is \(X = V(x^4 + y^4 + z^4 + w^4) \subseteq \PP^3_\CC\).
\(X\) is smooth: the partial derivatives \(4x^3, 4y^3, 4z^3, 4w^3\) vanish simultaneously only at the origin.
\(X\) is a K3 surface. By adjunction \(\omega_X \cong \OO_X(4 - 4) = \OO_X\), and the sequence \(0 \to \OO_{\PP^3}(-4) \to \OO_{\PP^3} \to \OO_X \to 0\) gives \(H^1(X, \OO_X) = 0\) because \(H^1(\OO_{\PP^3}) = H^2(\OO_{\PP^3}(-4)) = 0\).
Its invariants are \(\chi(\OO_X) = 2\), topological Euler characteristic \(24\) by Noether’s formula with \(K_X^2 = 0\), \(b_2 = 22\) and \(h^{1,1} = 20\).
\(X\) contains exactly \(48\) lines. If \(\zeta^4 = \eta^4 = -1\), the line \(\{x = \zeta y,\ z = \eta w\}\) lies on \(X\), since \(x^4 + y^4 = (\zeta^4 + 1) y^4 = 0\) and likewise for \(z, w\). There are \(16\) such lines for each of the \(3\) ways to split the coordinates into two pairs. Each line \(L\) has \(L^2 = -2\) by adjunction.
The lines span a sublattice of rank \(20\) in \(\operatorname{NS}(X)\), so \(X\) has Picard number \(20\), the maximum for a complex K3 surface.