The genus in its several senses (Tag D-G1AEH)
For a projective integral curve \(C\) over \(k = \bar{k}\):
the arithmetic genus is \(p_a(C) = 1 - \chi(\OO_C)\), the constant term of the Hilbert polynomial read with a sign;
the geometric genus is \(p_g(C) = h^0(\tilde{C}, \omega_{\tilde{C}}) = p_a(\tilde{C})\), computed on the normalization \(\tilde{C}\).
(One genus for a nonsingular curve)
If \(X\) is a nonsingular projective curve over \(k = \bar k\), then \[ p_a(X) = p_g(X) = \dim_k H^1(X, \OO_X) , \] the genus \(g\) of \(X\): since \(h^0(\OO_X) = 1\), \(p_a = 1 - \chi(\OO_X) = h^1(\OO_X)\), and Serre duality gives \(h^0(\omega_X) = h^1(\OO_X)\). [@Har10a, Exercise III.5.3]
\(p_g \leq p_a\), with \[ p_a(C) - p_g(C) = \sum_{p \in C} \delta_p , \] the sum of the delta invariants \(\delta_p = \dim_k (\nu_* \OO_{\tilde{C}} / \OO_C)_p\) over the singular points. A node contributes \(\delta = 1\), an ordinary cusp \(\delta = 1\), an ordinary \(m\)-fold point \(\binom{m}{2}\).
The normalization formula gives \(0\le p_g(C)\le p_a(C)\), with equality of the two genera exactly when \(C\) is smooth. A plane curve of degree \(d\) has \(p_a = \binom{d-1}{2}\) for every choice of singularities, and \(p_g=\binom{d-1}{2}-\sum_p\delta_p\); a nodal cubic has \(p_a = 1\) and \(p_g = 0\).
Both genera are invariants of the abstract curve: \(p_a = 1 - \chi(\OO_C)\), and \(p_g\) is computed on the normalization. The degree depends on the embedding. For a nonsingular projective complex curve, both genera equal half its first Betti number, by the Hodge decomposition and Serre duality [@Har10a, Appendix B and Exercise III.5.3]. For a singular complex curve, the topological genus of its smooth model is \(p_g(C)\); a nodal cubic has smooth model \(\PP^1\), of genus zero, and arithmetic genus one.
The arithmetic genus is read from the Hilbert polynomial, the geometric genus is the genus of the normalization, and over \(\CC\) the topological genus of the smooth model equals the geometric genus. For an integral projective curve \(C\) over an algebraically closed field with normalization \(\tilde C\), \(p_a(C)=g(\tilde C)+\sum_{p\in C}\delta_p\), where \(\delta_p=\dim_k\tilde\OO_p/\OO_{C,p}\) and \(\tilde\OO_p\) is the integral closure of \(\OO_{C,p}\); so \(p_a(C)=g(\tilde C)\) exactly when \(C\) is smooth.
Both genera are invariants of the curve, while the degree depends on the embedding: the twisted cubic and a line in \(\PP^3\) are both isomorphic to \(\PP^1\), of genus \(0\), and have degrees \(3\) and \(1\).
Computing the genus
Plane curves. A plane curve of degree \(d\) has \(p_a=\binom{d-1}{2}\), and its geometric genus is \(p_a-\sum_p\delta_p\).
Hilbert polynomial. A curve \(C\subseteq\PP^n\) of degree \(d\) has Hilbert polynomial \(P_C(m)=dm+1-p_a\).
Maps to a known curve. A finite separable morphism \(f\colon X\to Y\) of smooth projective curves satisfies Riemann–Hurwitz.
Riemann--Hurwitz (Tag T-LKT0U)
Let \(f : X \to Y\) be a finite separable morphism of smooth projective curves of degree \(n\). Then \[ 2g_X - 2 = n(2g_Y - 2) + \deg R , \] where \(R = \sum_{p} \length(\Omega_{X/Y})_p\) is the ramification divisor. When every ramification is tame, \(\deg R = \sum_p (e_p - 1)\).
(Proof structure)
The relative cotangent sequence for \(X \to Y \to \Spec k\) is \[ f^* \Omega_{Y} \to \Omega_{X} \to \Omega_{X/Y} \to 0 . \] For a nonconstant separable map of smooth curves the left map is injective, so the sequence is short exact, and \(\Omega_{X/Y}\) is a torsion sheaf supported at the ramification points. Taking degrees gives \(\deg \Omega_X = n \deg \Omega_Y + \deg R\), and \(\deg \Omega_C = 2g_C - 2\) from Riemann–Roch.
Thus the proof reduces to injectivity of the map on differentials, exactness of the cotangent sequence, and taking degrees. Separability is what makes the left map injective, and in characteristic \(p\) the Frobenius \(\PP^1 \to \PP^1\) shows the statement fails without it — the map on differentials is zero.
(Using it)
The formula is used in both directions. Forward, with \(Y = \PP^1\): a degree-\(2\) cover of \(\PP^1\) branched at \(2g+2\) points has genus \(g\), which is every hyperelliptic curve. Backward: a curve admitting a map of known degree to a known curve has its genus pinned, and since \(\deg R \geq 0\), a nonconstant map \(X \to Y\) forces \(g_X \geq g_Y\) — no curve maps onto one of larger genus.
The ramification divisor \(R\) is the divisor of zeros of \(f^*\colon f^*\Omega_Y\to\Omega_X\), so \(K_X\sim f^*K_Y+R\); taking degrees gives the Riemann–Hurwitz formula.
The small genera
Curves of genus $0$ (Tag PR-VGA2L)
Let \(C\) be a smooth projective curve of genus \(0\) over a field \(k\).
If \(C\) has a \(k\)-rational point, then \(C \cong \PP^1_k\).
In general \(C\) embeds as a smooth conic in \(\PP^2_k\) by the anticanonical system \(\abs{-K}\), which has degree \(2\) and \(\ell(-K) = 3\).
Over \(k = \bar{k}\), and over any finite field, the first case always holds.
Riemann–Roch does the work. With \(g = 0\) and \(\deg D = 1\), \(\ell(D) = 2\) and \(\ell(K - D) = 0\) since \(\deg(K-D) = -3 < 0\), so a degree-one divisor gives a degree-one map to \(\PP^1\), which is an isomorphism. A \(k\)-point is exactly such a divisor, and that is where the rationality hypothesis enters.
Without a rational point the anticanonical divisor still has degree \(2\) and \(\ell = 3\), giving the conic. The conic \(x^2 + y^2 + z^2 = 0\) over \(\RR\) is the standard curve of genus zero that is not \(\PP^1\).
Over a finite field the second case cannot occur: a smooth conic over \(\FF_q\) always has a rational point by the Chevalley–Warning theorem, so every genus-zero curve there is \(\PP^1\). Thus over a finite field every smooth projective genus-zero curve has a rational point and is isomorphic to \(\PP^1\).
Twisted forms of the line
Severi--Brauer varieties, and the Brauer class of a twisted projective space (Tag D-VARSEVBRAUER)
Let \(k\) be a field with separable closure \(k^s\). A Severi–Brauer variety of dimension \(n\) over \(k\) is a variety \(X\) over \(k\) with \(X_{k^s} \cong \PP^n_{k^s}\).
Isomorphism classes of Severi–Brauer varieties of dimension \(n\) over \(k\) are in bijection with \(H^1(\Gal(k^s/k), \PGL_{n+1}(k^s))\), hence with isomorphism classes of central simple \(k\)-algebras of degree \(n+1\); each has a class in \(\operatorname{Br}(k)\).
(Châtelet) \(X \cong \PP^n_k\) if and only if \(X(k) \neq \emptyset\), if and only if its Brauer class is trivial.
The Severi–Brauer varieties of dimension \(1\) are the smooth conics. The conic \(x^2 + y^2 + z^2 = 0\) over \(\RR\) has no real point, so it is not \(\PP^1_\RR\); it corresponds to the Hamilton quaternions. If \(k\) is finite, or is the function field of a curve over an algebraically closed field, then \(\operatorname{Br}(k) = 0\) (Wedderburn, Tsen), so every Severi–Brauer variety over \(k\) is a projective space.
A family of genus-zero curves over a curve is a $\PP^1$-bundle (Tag P-AGXMISCTSENPONEBUNDLE)
Use Tsen’s theorem to show that given a flat family \(X\to Y\) with \(Y\) a curve where the fibers smooth curves of genus 0, this determines a Zariski \(\PP^1\)-bundle iff there exists a relative degree 1 line bundle on \(X\) over \(Y\).
Solution
Let \(X\) and \(Y\) be integral schemes, let \(Y\) be a smooth curve over an algebraically closed field \(k\), and let \[ f:X\longrightarrow Y \] be projective and flat with smooth genus-zero fibres. Here a Zariski \(\PP^1\)-bundle means a projective bundle \(\PP_Y(\mce)\) for a rank-two locally free sheaf \(\mce\). A relative degree-one line bundle is an invertible sheaf \(\mcl\) on \(X\) such that \[ \deg(\mcl|_{X_y})=1 \] for every \(y\in Y\). Since a projective morphism is of finite presentation, flatness together with smooth geometric fibres implies that \(f\) is smooth.
<1>1. If \[ X\cong\PP_Y(\mce) \] for a rank-two vector bundle \(\mce\) on \(Y\), then \(X\) has a relative degree-one line bundle.
The projective bundle carries its tautological quotient line bundle \[ \OO_{\PP_Y(\mce)}(1). \] On every fibre \[ \PP(\mce_y)\cong\PP^1_{\kappa(y)}, \] its restriction is \(\OO_{\PP^1}(1)\), which has degree \(1\).
<1>2. Conversely, suppose \(\mcl\) is a relative degree-one line bundle. For every fibre \(C=X_y\), \[ h^0(C,\mcl_y)=2, \qquad h^1(C,\mcl_y)=0. \]
The curve \(C\) is smooth proper of genus \(0\), and \[ \deg\mcl_y=1. \] Riemann–Roch gives \[ h^0(C,\mcl_y)-h^1(C,\mcl_y) = \deg\mcl_y+1-g =2. \] By Serre duality, \[ h^1(C,\mcl_y) = h^0(C,\omega_C\tensor\mcl_y^{-1}). \] The line bundle on the right has degree \[ -2-1=-3, \] so it has no nonzero section. Hence \(h^1=0\) and \(h^0=2\).
<1>3. The sheaf \[ \mce=f_*\mcl \] is locally free of rank \(2\), and its formation commutes with base change: \[ \mce\tensor\kappa(y) \cong H^0(X_y,\mcl_y). \]
The line bundle \(\mcl\) is flat over \(Y\) because it is locally free over \(\OO_X\) and \(X\) is flat over \(Y\). By step <1>2, the fibre dimensions \[ h^0(X_y,\mcl_y)=2 \] are constant. Cohomology and base change T-COHBC therefore gives that \(f_*\mcl\) is locally free of rank \(2\) and commutes with restriction to every fibre.
<1>4. For every fibre \(C=X_y\), the line bundle \(\mcl_y\) is isomorphic to \(\OO_{\PP^1}(1)\) after identifying \(C\cong\PP^1_{\kappa(y)}\), and its complete linear system defines an isomorphism \[ C\xrightarrow{\sim}\PP\bigl(H^0(C,\mcl_y)\bigr). \]
By step <1>2, \(\mcl_y\) has a nonzero section. Its zero divisor is an effective divisor of degree \(1\), hence a single \(\kappa(y)\)-rational point \(p\). A smooth proper genus-zero curve with a rational point is \(\PP^1_{\kappa(y)}\) D-VARSEVBRAUER, and under such an identification \[ \mcl_y\cong\OO_{\PP^1}(p)\cong\OO_{\PP^1}(1). \] The complete linear system of \(\OO_{\PP^1}(1)\) is the standard isomorphism to projective one-space.
<1>5. The evaluation homomorphism \[ f^*\mce\longrightarrow\mcl \] is surjective and defines a \(Y\)-morphism \[ g:X\longrightarrow\PP_Y(\mce) \] whose restriction to every fibre is an isomorphism.
By step <1>3, restriction of the evaluation map to \(X_y\) is \[ H^0(X_y,\mcl_y)\tensor\OO_{X_y} \longrightarrow \mcl_y. \] By step <1>4 this is the usual evaluation map for \(\OO_{\PP^1}(1)\), so it is surjective. The cokernel of the global evaluation map is coherent and has zero restriction to every fibre; Nakayama’s lemma therefore implies that the cokernel is zero.
A quotient \[ f^*\mce\twoheadrightarrow\mcl \] defines the displayed morphism to the projective bundle. Base change in step <1>3 identifies its fibre map with the complete-linear-system map in step <1>4, hence \[ g_y:X_y\xrightarrow{\sim}\PP(\mce_y) \] for every \(y\).
<1>6. The morphism \(g\) is an isomorphism. Thus a relative degree-one line bundle makes \(X\) a Zariski \(\PP^1\)-bundle.
The morphism \(g\) is proper because \(X\) is proper over \(Y\) and \(\PP_Y(\mce)\) is separated over \(Y\). Since every fibre map \(g_y\) is an isomorphism, \(g\) is quasi-finite. Proper and quasi-finite implies finite.
The map on the generic fibre is an isomorphism by step <1>5, so \(g\) is birational. Since \(Y\) is smooth and \(\mce\) is locally free, \[ \PP_Y(\mce) \] is smooth over the regular curve \(Y\), hence regular and in particular normal. A finite birational morphism onto a normal integral scheme is an isomorphism. Therefore \[ g:X\xrightarrow{\sim}\PP_Y(\mce). \]
<1>7. Hence \[ \boxed{ X\cong\PP_Y(\mce)\text{ for a rank-two }\mce \iff X\text{ carries a relative degree-one line bundle}. } \]
Step <1>1 proves the forward implication and steps <1>2–<1>6 prove the reverse implication.
<1>8. Tsen’s theorem supplies such a degree-one line bundle for the generic genus-zero fibre.
Let \[ K=k(Y) \] and let \(X_\eta\) be the generic fibre. It is a smooth proper genus-zero curve over \(K\), hence a one-dimensional Severi–Brauer variety. Tsen’s theorem gives \[ \operatorname{Br}(K)=0, \] so D-VARSEVBRAUER yields \[ X_\eta\cong\PP^1_K. \] In particular \(X_\eta\) has a \(K\)-rational point, which is a rational section \[ s_\eta:\eta\dashrightarrow X \] of \(f\).
For a closed point \(y\in Y\), the local ring \(\OO_{Y,y}\) is a DVR with fraction field \(K\). Properness of \(f\) and the valuative criterion extend \(s_\eta\) uniquely over \(\Spec\OO_{Y,y}\). Since \(X\to Y\) is of finite presentation, this morphism from the local scheme descends to a section over some open neighborhood of \(y\). The resulting local sections all agree on the generic point; separatedness of \(f\) makes them agree on their overlaps. They therefore glue to a global section \[ s:Y\longrightarrow X. \] Since \(f\) is a smooth relative curve, its section \(s\) is a regular closed immersion of codimension one. Thus \(s(Y)\) is an effective Cartier divisor meeting every fibre in one point. Therefore \[ \OO_X(s(Y)) \] has relative degree \(1\).
<1>9. In particular, over a smooth curve \(Y\) over an algebraically closed field, every such smooth genus-zero family is a Zariski \(\PP^1\)-bundle.
Step <1>8 constructs a relative degree-one line bundle, and step <1>7 applies the criterion.
<1>10. Q.E.D.
Step <1>7 proves the requested equivalence, and steps <1>8–<1>9 use Tsen’s theorem to produce the relative degree-one line bundle.