Let \(X\) be an integral projective scheme of dimension 1 over \(k\). Let \(X_{\reg}\) be the set of regular points of \(X\).
Let \(D=\sum n_i P_i\) be a divisor with support in \(X_{\reg}\), i.e., all \(P_i \in X_{\reg}\). Then define \(\deg D=\sum n_i\). Let \(\mcl(D)\) be the associated invertible sheaf on \(X\), and show that \[ \chi(\mcl(D))=\deg D+1-p_a . \]
Show that any Cartier divisor on \(X\) is the difference of two very ample Cartier divisors. Use (II, Ex. 7.5).
Conclude that every invertible sheaf \(\mcl\) on \(X\) is isomorphic to \(\mcl(D)\) for some divisor \(D\) with support in \(X_{\reg}\).
Assume furthermore that \(X\) is a locally complete intersection in some projective space. Then by (III, 7.11) the dualizing sheaf \(\omega_X\) is an invertible sheaf on \(X\), so we can define the canonical divisor \(K\) to be a divisor with support in \(X_{\reg}\) corresponding to \(\omega_X\). Then the formula of a. becomes \[ l(D)-l(K-D)=\deg D+1-p_a \]
Solution
1. If \(P\in X_{\reg}\) and \(D\) is supported in \(X_{\reg}\), then\[ \chi(\mcl(D+P)) = \chi(\mcl(D))+1. \]
Because \(P\) is regular on the one-dimensional scheme \(X\), it is an effective Cartier divisor. Tensoring \[ 0 \longrightarrow \mco_X(-P) \longrightarrow \mco_X \longrightarrow k(P) \longrightarrow 0 \] by \(\mcl(D+P)\) gives \[ 0 \longrightarrow \mcl(D) \longrightarrow \mcl(D+P) \longrightarrow k(P) \longrightarrow 0. \] Since \(k\) is algebraically closed, \[ \chi(k(P))=1. \] Additivity of Euler characteristic in a short exact sequence therefore gives the formula.
2. For every divisor\[ D=\sum_i n_iP_i \]supported in \(X_{\reg}\),\[ \boxed{ \chi(\mcl(D)) = \deg D+1-p_a(X). } \]
Starting from \(D=0\), repeatedly apply step 1 when increasing one coefficient by \(1\). The same formula read backwards applies when decreasing a coefficient by \(1\). Hence \[ \chi(\mcl(D)) = \chi(\mco_X)+\sum_i n_i. \] By definition of the arithmetic genus of a projective integral curve, \[ \chi(\mco_X)=1-p_a(X), \] while \[ \sum_i n_i=\deg D. \] This proves part (a).
3. Let \(C\) be any Cartier divisor on \(X\). Then\[ C=A-B \]for two very ample Cartier divisors \(A\) and \(B\).
Choose a very ample Cartier divisor \(H\) on the projective curve \(X\). The invertible sheaf \[ \mco_X(H) \] is ample and globally generated.
By Exercise II.7.5, for all sufficiently large \(n\) the sheaf \[ \mco_X(C+(n-1)H) \] is globally generated. For such an \(n\), part (d) of that exercise says that \[ \mco_X(C+nH) = \mco_X(H)\tensor\mco_X(C+(n-1)H) \] is very ample.
Also \[ \mco_X(nH) = \mco_X(H)\tensor\mco_X((n-1)H) \] is very ample, because \(\mco_X((n-1)H)\) is globally generated. Thus \[ A=C+nH, \qquad B=nH \] are very ample Cartier divisors and \[ C=A-B. \] This proves part (b).
4. Every invertible sheaf on \(X\) is isomorphic to \(\mcl(D)\) for adivisor \(D\) supported in \(X_{\reg}\).
Let \(\mcl\) be invertible. Since \(X\) is integral, a nonzero rational trivialization of \(\mcl\) gives a Cartier divisor \(C\) with \[ \mcl\cong\mcl(C). \] By step 3, write \[ C=A-B \] with \(A\) and \(B\) very ample.
The singular locus of the integral curve \(X\) is finite. The complete linear systems \(\abs{A}\) and \(\abs{B}\) are base-point free. Therefore one may choose effective divisors \[ A'\sim A, \qquad B'\sim B \] whose supports avoid every singular point: under the embeddings defined by \(A\) and \(B\), choose hyperplanes avoiding the finite images of the singular points.
Then \[ D=A'-B' \] is supported in \(X_{\reg}\) and \[ \mcl(D) \cong \mcl(A-B) \cong \mcl(C) \cong \mcl. \] This proves part (c).
5. Assume \(X\) is a locally complete intersection in projective space.Then there is a divisor \(K\) supported in \(X_{\reg}\) such that\[ \mcl(K)\cong\omega_X. \]
For a projective locally complete intersection curve, the dualizing sheaf \(\omega_X\) is invertible. Apply part (c), proved in step 4, to this invertible sheaf. It gives a divisor \(K\) supported in \(X_{\reg}\) with \[ \mcl(K)\cong\omega_X. \] This is the canonical divisor specified in the statement.
6. For every divisor \(D\) supported in \(X_{\reg}\),\[ H^1(X,\mcl(D))^\vee \cong H^0(X,\mcl(K-D)). \]
Serre duality for the projective Cohen–Macaulay curve \(X\) gives \[ H^1(X,\mcl(D))^\vee \cong \Hom_X(\mcl(D),\omega_X). \] Since \(\mcl(D)\) is invertible and step 5 identifies \(\omega_X\cong\mcl(K)\), \[ \Hom_X(\mcl(D),\omega_X) \cong H^0\bigl(X,\omega_X\tensor\mcl(-D)\bigr) \cong H^0(X,\mcl(K-D)). \]
7. The singular-curve Riemann–Roch formula is\[ \boxed{ \ell(D)-\ell(K-D) = \deg D+1-p_a(X). } \]
By step 6, \[ h^1(X,\mcl(D)) = \ell(K-D). \] Therefore \[ \chi(\mcl(D)) = \ell(D)-\ell(K-D). \] Substitute the Euler-characteristic formula from step 2: \[ \chi(\mcl(D)) = \deg D+1-p_a(X). \] This is exactly the displayed formula in part (d).
8. Q.E.D.
Step 2 proves part (a), step 3 proves part (b), step 4 proves part (c), and steps <1>5–<1>7 prove part (d).