## Return to Answer

4 Added: Complement

Dear Hugo, the wonderful formalism of schemes allows us to have a Riemann-Roch theorem for a projective curve $X$ over an arbitrary field $k$, even without any assumption of smoothness. It says, like in the good old times of Riemann surfaces, that for a Cartier divisor $D$ on $X$ we have $$\chi (\mathcal O_X(D))= deg(D)+ \chi (\mathcal O_X)$$

There is a dualizing sheaf $\omega$ and Serre duality yields the formula $$h^0(X,\mathcal O_X(D))-h^0(X,\omega \otimes\mathcal O_X(-D))=1-p_a(X)+deg D$$ where $p_a(X)=1-\chi(\mathcal O_x)$ is the so called arithmetic genus of the curve.

Everything is in our friend Qing Liu's fantastic book Algebraic Geometry and Arithmetic Curves but I bet he's too modest to give you this obvious answer!

Edit The first displayed formula is actually valid in even greater generality: it holds for any projective curve $X$ (smooth or not) over an arbitrary artinian ring $k$. The proof is on page 164 of

Altman, A.; Kleiman, S., Introduction to Grothendieck duality theory. Lecture Notes in Mathematics No. 146, Springer-Verlag, Berlin-New York, 1970

Complement As an answer to Hugo's question in his comment below, let me add that indeed, in the case of a smooth projective curve over a field $k$, the dualizing sheaf $\omega$ is nothing else than the canonical sheaf . More generally for a smooth projective variety of dimension $r$ over $k$, the dualizing sheaf is just the canonical sheaf $\omega=\Omega^r_{X/k}$. This is (a special case of) Theorem I.4.6, page 14 in Altman-Kleiman's monograph.

3 suppressed redundant word "book" that I had used twice

Dear Hugo, the wonderful formalism of schemes allows us to have a Riemann-Roch theorem for a projective curve $X$ over an arbitrary field $k$, even without any assumption of smoothness. It says, like in the good old times of Riemann surfaces, that for a Cartier divisor $D$ on $X$ we have $$\chi (\mathcal O_X(D))= deg(D)+ \chi (\mathcal O_X)$$

There is a dualizing sheaf $\omega$ and Serre duality yields the formula $$h^0(X,\mathcal O_X(D))-h^0(X,\omega \otimes\mathcal O_X(-D))=1-p_a(X)+deg D$$ where $p_a(X)=1-\chi(\mathcal O_x)$ is the so called arithmetic genus of the curve.

Everything is in our friend Qing Liu's book fantastic book Algebraic Geometry and Arithmetic Curves but I bet he is he's too modest to give you this obvious answer!

Edit The first displayed formula is actually valid in even greater generality: it holds for any projective curve $X$ (smooth or not) over an arbitrary artinian ring $k$. The proof is on page 164 of

Altman, A.; Kleiman, S., Introduction to Grothendieck duality theory. Lecture Notes in Mathematics No. 146, Springer-Verlag, Berlin-New York, 1970

2 added reference for statement over artin ring

Dear Hugo, the wonderful formalism of schemes allows us to have a Riemann-Roch theorem for a projective curve $X$ over an arbitrary field $k$, even without any assumption of smoothness. It says, like in the good old times of Riemann surfaces, that for a Cartier divisor $D$ on $X$ we have $$\chi (\mathcal O_X(D))= deg(D)+ \chi (\mathcal O_X)$$

There is a dualizing sheaf $\omega$ and Serre duality yields the formula $$h^0(X,\mathcal O_X(D))-h^0(X,\omega \otimes\mathcal O_X(-D))=1-p_a(X)+deg D$$ where $p_a(X)=1-\chi(\mathcal O_x)$ is the so called arithmetic genus of the curve.

Everything is in our friend Qing Liu's book fantastic book Algebraic Geometry and Arithmetic Curves but I bet he is too modest to give you this obvious answer!

Edit The first displayed formula is actually valid in even greater generality: it holds for any projective curve $X$ (smooth or not) over an arbitrary artinian ring $k$. The proof is on page 164 of

Altman, A.; Kleiman, S., Introduction to Grothendieck duality theory. Lecture Notes in Mathematics No. 146, Springer-Verlag, Berlin-New York, 1970

1

Dear Hugo, the wonderful formalism of schemes allows us to have a Riemann-Roch theorem for a projective curve $X$ over an arbitrary field $k$, even without any assumption of smoothness. It says, like in the good old times of Riemann surfaces, that for a Cartier divisor $D$ on $X$ we have $$\chi (\mathcal O_X(D))= deg(D)+ \chi (\mathcal O_X)$$

There is a dualizing sheaf $\omega$ and Serre duality yields the formula $$h^0(X,\mathcal O_X(D))-h^0(X,\omega \otimes\mathcal O_X(-D))=1-p_a(X)+deg D$$ where $p_a(X)=1-\chi(\mathcal O_x)$ is the so called arithmetic genus of the curve.

Everything is in our friend Qing Liu's book fantastic book Algebraic Geometry and Arithmetic Curves but I bet he is too modest to give you this obvious answer!
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