Over an algebraically closed field \(k\) of characteristic \(p>0\), a finite morphism \(X\to Y\) of smooth projective curves factors as a purely inseparable morphism \(X\to X'\) followed by a separable morphism \(X'\to Y\), and every purely inseparable morphism of curves is a composite of \(k\)-linear Frobenius morphisms.
Frobenius as a morphism over \(k\)
The $k$-linear Frobenius and the Frobenius twist (Tag D-IV2FROBTWIST)
Let \(X\) be a scheme all of whose local rings contain \(\FF_p\). The Frobenius morphism \(F : X \to X\) is the identity on the underlying topological space, with \[ F^\sharp : \OO_X \to \OO_X, \qquad f \mapsto f^p . \] This is a ring homomorphism in characteristic \(p\), so \(F\) is a morphism of schemes.
(The twist and the $k$-linear Frobenius)
For \(X\) over \(k\) with structure morphism \(\pi : X \to \Spec k\), let \(X_p\) denote \(X\) with the structure morphism twisted by Frobenius, so that \(k\) acts on \(\OO_{X_p}\) through \(p\)-th powers. Then \(F\) becomes a \(k\)-morphism \[ F' : X_p \to X , \] the \(k\)-linear Frobenius. For \(X\) a curve over perfect \(k\) it is finite of degree \(p\), and on function fields it is the inclusion \[ k(X) \subseteq k(X)^{1/p} . \]
(Effect of the twist)
The absolute \(F\) is not a morphism over \(k\) unless \(k=\FF_p\): \(F^\sharp\) sends \(\lambda \in k\) to \(\lambda^p\), so \(\pi\circ F=\operatorname{Frob}_{\Spec k}\circ\pi\), where \(\operatorname{Frob}_{\Spec k}\) is the Frobenius of \(\Spec k\). Over \(k=\FF_p\) the Frobenius of \(\Spec k\) is the identity, so \(F=F'\). In \(X_p\) the \(k\)-structure is composed with the \(p\)-th power map of \(k\), so \(\pi\circ F'\) is the structure morphism of \(X_p\) and \(F'\) is a morphism over \(\Spec k\).
The extension \(k(X)^{1/p}/k(X)\) has degree \(p\): \(k(X)\) has transcendence degree \(1\) over the perfect field \(k\), so \(k(X)^{1/p}\) is generated over \(k(X)\) by the \(p\)-th root of a separating transcendental element. For \(X=\PP^1\), this is \(k(t) \subseteq k(t^{1/p})\), of degree \(p\) since \(t^{1/p}\) is a root of the irreducible polynomial \(T^p-t\).
Twisting applies the Frobenius of \(k\) to the coefficients of the defining equations, so \(X_p \cong X\) over \(k\) whenever \(X\) is defined over \(\FF_p\), for instance \(X = \AA^1\). For \(k\) algebraically closed and an elliptic curve \(E\) over \(k\), the twist replaces \(j(E)\) by a \(p\)-th power or \(p\)-th root of it, so \(E_p \cong E\) over \(k\) if and only if \(j(E)^p=j(E)\), that is, \(j(E) \in \FF_p\); a curve with \(j(E)\notin\FF_p\) has \(E_p\not\cong E\).
(Absolute versus relative Frobenius)
The absolute Frobenius of \(\PP^n\) is finite, flat, bijective, and nowhere smooth, because \(d(t^p) = 0\) (FE-MORFROB). The same identity \(d(t^p)=0\) shows that \(k(X)^{1/p}/k(X)\) is purely inseparable, which PR-IV2INSEP uses.
The absolute Frobenius is a morphism of \(k\)-schemes only when \(a^p=a\) for every \(a\in k\); the Frobenius twist modifies the structure morphism of the source so that the relative Frobenius is \(k\)-linear. The \(k\)-linear Frobenius of a curve is finite of degree \(p\) and induces \(k(X)\subseteq k(X)^{1/p}\), an extension of degree \(p\) because \(k(X)\) has transcendence degree \(1\) over the perfect field \(k\).
Purely inseparable morphisms of curves are composites of Frobenius (Tag PR-IV2INSEP)
Let \(f : X \to Y\) be a finite morphism of curves over \(k = \kbar\) with \(\characteristic k = p\), and suppose \(k(X)/k(Y)\) is purely inseparable of degree \(p^n\). Then \(X\) and \(Y\) are isomorphic as schemes, \(g(X) = g(Y)\), and \(f\) is the \(n\)-fold composite of \(k\)-linear Frobenius morphisms. [@Har10a]
(Why the statement is this strong)
Purely inseparable means every element of \(k(X)\) has a \(p\)-power lying in \(k(Y)\), so \(k(Y) \supseteq k(X)^{p^n}\) and the extension is squeezed between \(k(X)\) and \(k(X)^{p^n}\). A purely inseparable extension of degree \(p^n\) of a function field of transcendence degree \(1\) over a perfect field is therefore \(k(X) \subseteq k(X)^{1/p^n}\) up to isomorphism, and that inclusion is exactly the one realised by the \(n\)-fold Frobenius of D-IV2FROBTWIST. The source of the \(k\)-linear Frobenius \(X_p \to X\) is \(X\) itself with its structure morphism composed with the Frobenius of \(\Spec k\), so \(X \cong Y\) as schemes, and the genus, computed from \(\dim_k H^1(\OO)\) on that one scheme, cannot move. The Frobenius morphism itself is a homeomorphism but not an isomorphism of schemes: on \(\Spec k[t]\) it is given by \(t \mapsto t^p\), and \(k[t] \to k[t]\), \(f \mapsto f^p\), is not surjective.
The isomorphism is of schemes, not of \(k\)-schemes, and that distinction is the whole content. What changes under the twist is the \(k\)-structure, not the space and not the sheaf of rings up to isomorphism, so every invariant computed after forgetting \(k\) is preserved and the morphism is still not an isomorphism over \(k\) and still has degree \(p^n\).
(The trap)
Riemann–Hurwitz does not apply and cannot be repaired here. An inseparable \(f\) kills differentials, \(f^* \Omega_Y \to \Omega_X\) is the zero map, so the cotangent sequence is not left exact and the degree count that proves the formula has nothing to count. Concretely \(\Omega_{X/Y} \cong \Omega_X\), a line bundle of degree \(2g-2\) rather than a torsion sheaf, so the “ramification divisor” is not even the right kind of object.
The use of this proposition is as the missing half of the genus inequality. Every finite morphism of curves factors as a separable morphism after a purely inseparable one; Riemann–Hurwitz gives \(g \geq\) across the separable part and this proposition gives equality across the inseparable part. That is how \(g(X) \geq g(Y)\) holds for every finite morphism in characteristic \(p\), which is the step T-IV2LUROTH needs.
The source and target of a purely inseparable morphism of curves are isomorphic as schemes, not over \(k\), so they have the same genus. For the \(k\)-linear Frobenius \(F\colon X\to X'\), the map \(F^*\Omega_{X'}\to\Omega_X\) is zero, so \(\Omega_{X/X'}\cong\Omega_X\) is a line bundle, and the Riemann–Hurwitz formula \(2g-2=p(2g-2)\) fails for \(g\neq1\).
Lüroth’s theorem
Lüroth's theorem, and the Lüroth problem in higher dimension (Tag T-IV2LUROTH)
Let \(k = \kbar\) and let \(L\) be a field with \(k \subsetneq L \subseteq k(t)\). Then \(L\) is purely transcendental: \(L = k(u)\) for some \(u\).
\(L\) is a subfield of a finitely generated field of transcendence degree \(1\), so \(\trdeg_k L \leq 1\), and \(\trdeg_k L = 1\) because \(L \neq k\) and \(k\) is algebraically closed, so any element of \(L \setminus k\) is transcendental over \(k\). Such an \(L\) is itself finitely generated over \(k\), hence \(L = k(Y)\) for a unique smooth projective curve \(Y\).
The inclusion \(L \subseteq k(t) = k(\PP^1)\) is a finite extension of function fields, so it is induced by a finite morphism \(f : \PP^1 \to Y\). A finite morphism of curves never decreases the genus, so \(g(Y) \leq g(\PP^1) = 0\) and \(g(Y) = 0\). A genus-\(0\) curve over \(k = \kbar\) is \(\PP^1\), so \(Y \cong \PP^1\) and \(L \cong k(u)\).
(What the proof actually rests on)
The one input is \(g(X) \geq g(Y)\) for a finite morphism \(X \to Y\) of curves, and that is Riemann–Hurwitz read backwards: for separable \(f\) of degree \(n\), \[ 2 g_X - 2 = n(2 g_Y - 2) + \deg R, \qquad \deg R \geq 0 , \] so \(g_X < g_Y\) is impossible. The inseparable case is not an exception but a separate mechanism: factor \(f\) through its purely inseparable part, which preserves the genus, and apply the above to the separable part. This is the role of the inseparable factorization in PR-IV2INSEP.
Algebraic closedness enters twice. It makes every element of \(L \setminus k\) transcendental, and it turns \(g(Y) = 0\) into \(Y \cong \PP^1\) — over a general field a genus-\(0\) curve is a conic, which is \(\PP^1\) only when it has a rational point. The purely field-theoretic Lüroth theorem needs neither: over any field \(k\), every intermediate field \(k \subsetneq L \subseteq k(t)\) is \(k(u)\). The geometric proof above buys the statement back with the curve theory instead of with field theory.
(The Lüroth problem)
Restated as geometry, Lüroth says: a curve dominated by \(\PP^1\) is rational, that is, unirational implies rational in dimension \(1\). Asking the same in higher dimension is the Lüroth problem, and the answer splits by dimension and by characteristic.
In dimension \(2\) over \(k = \kbar\) of characteristic \(0\), unirational implies rational. The proof runs through Castelnuovo’s criterion, that \(q=P_2=0\) forces rationality; the characteristic hypothesis enters in deducing these vanishings from unirationality. Castelnuovo’s criterion itself holds in every characteristic, by Zariski and Mumford. What fails in characteristic \(p\) is the other half: a unirational surface need not satisfy \(q = P_2 = 0\). Zariski’s surfaces, purely inseparable covers of \(\PP^2\), are unirational and not rational, so the dimension-\(2\) Lüroth statement is false as soon as \(p > 0\) even though the criterion it invokes is not.
In dimension \(3\) it is false even over \(\CC\). The three standard counterexamples are the smooth cubic threefold, non-rational because its intermediate Jacobian is not a Jacobian of a curve; the smooth quartic threefold, non-rational because its group of birational self-maps is too small; and the Artin–Mumford double solid, non-rational because it has torsion in \(H^3\), which is a birational invariant. All three are unirational. Each counterexample is distinguished by a specific birational obstruction: the intermediate Jacobian, birational automorphism group, or torsion in \(H^3\).
In the proof, a finite morphism \(X\to Y\) of smooth projective curves satisfies \(g(X)\ge g(Y)\): Riemann–Hurwitz gives the inequality across the separable part, and Purely inseparable morphisms of curves are composites of Frobenius gives equality across the purely inseparable part. A genus-\(0\) curve over an algebraically closed field is \(\PP^1\).
Restated as geometry, Lüroth says unirational implies rational in dimension \(1\). In higher dimension this is the Lüroth problem: unirational implies rational for surfaces over an algebraically closed field of characteristic \(0\), and fails for surfaces in characteristic \(p\) (Zariski surfaces) and for threefolds over \(\CC\) (cubic threefolds, by the intermediate Jacobian).