Arithmetic genus, geometric genus and delta invariants

Setting. The normalization $\nu$ is a finite, affine and birational morphism of integral curves with the same function field [F1]; $X^{\mathrm{n}\mathrm{u}}$ is a smooth proper curve over the algebraically closed field $k$ and $g(X^{\mathrm{n}\mathrm{u}})=h^1(X^{\mathrm{n}\mathrm{u}},{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}})$ [F1]. In particular both $X$ and $X^{\mathrm{n}\mathrm{u}}$ are proper over $k$, so the Euler characteristic of [F3] is defined for all coherent sheaves occurring below.

The structure sequence. Since $\nu$ is birational and both sheaves sit inside the constant sheaf of the function field $k(X)$, the natural map ${\mathcal{O}}_X\to {\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}}$ is injective; let $\mathcal{Q}$ be its cokernel. Then $0⟶{\mathcal{O}}_X⟶{\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}}⟶\mathcal{Q}⟶0$ is a short exact sequence of coherent ${\mathcal{O}}_X$-modules: ${\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}}$ is coherent because $\nu$ is finite and proper [F2], and $\mathcal{Q}$ is a quotient of a coherent sheaf on the locally Noetherian scheme $X$.

Support and stalks. For a closed point $x$ one has ${\mathcal{Q}}_x=({\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}}{)}_x\mathrm{/}{\mathcal{O}}_{X,x}$ [F2], so ${\delta }_x(X)={\dim }_k{\mathcal{Q}}_x$, and ${\mathcal{Q}}_x=0$ exactly when $x$ is regular [F2]; the support $S=\{x:{\mathcal{Q}}_x\mathrm{\ne }0\}$ is the finite set of singular points of $X$ and ${\sum }_{x\in S}{\delta }_x(X)={\sum }_{x\in X}{\delta }_x(X)$.

Cohomology of the pushforward. Since $\nu$ is finite, hence affine, the comparison of [F5] identifies $H^q(X,{\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}})\cong H^q(X^{\mathrm{n}\mathrm{u}},{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}})$ for all $q\ge 0$; hence $\chi (X,{\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}})=\chi (X^{\mathrm{n}\mathrm{u}},{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}})$.

Euler characteristics of the structure sequence. Applying additivity [F4] to the sequence of step 1.2, all three terms being coherent on the proper curve $X$ [F3], gives $\chi (X,{\mathcal{O}}_X)=\chi (X,{\nu }_{\ast }{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}})-\chi (X,\mathcal{Q})$.

Cohomology of the delta quotient. Define the annihilator subsheaf $\mathcal{I}\subseteq {\mathcal{O}}_X$ by requiring a local function to act as the zero endomorphism of $\mathcal{Q}$; this is a sheaf ideal because vanishing of a sheaf morphism is local. On an affine open $U=\mathrm{Spec}A$, write $\mathcal{Q}{\mathrm{\mid }}_U\cong \tilde{M}$ with $M$ finitely generated [F8], and put $J_U={\mathrm{Ann}}_A(M)=\mathrm{\Gamma }(U,\mathcal{I})$; the equality holds because $a\in A$ annihilates $M$ exactly when it annihilates every localization $M_f=\mathcal{Q}(D(f))$ on the principal-open basis of $U$. These ideals localize correctly: if $M=0$ the equality ${\mathrm{Ann}}_{A_f}(M_f)=(J_U{)}_f$ is immediate; otherwise choose finite generators $m_1,\dots ,m_r$. If $a\mathrm{/}{f}^n$ annihilates $M_f$, then for each $j$ some $e_j\ge 0$ has $f^{e_j}a{m}_j=0$; taking $e={\max }_j{e}_j$ gives $f^{e}a\in J_U$, hence $a\mathrm{/}{f}^n\in (J_U{)}_f$. The reverse inclusion is immediate. By the definition of $\mathcal{I}$, $\mathrm{\Gamma }(D(f),\mathcal{I})={\mathrm{Ann}}_{A_f}(M_f)$, so this localization identity shows that $\mathcal{I}{\mathrm{\mid }}_U=\widetilde{J_U}$ on the principal-open basis. The affine descriptions agree on overlaps because they are restrictions of the intrinsic annihilator subsheaf; in particular $\mathcal{I}$ is quasi-coherent [F8]. By [F9], on each such $U$ the support of $\mathcal{Q}$ is $V(J_U)$, and the closed subscheme $i:S=V(\mathcal{I})\hookrightarrow X$ from [F10] therefore has underlying space exactly the finite set in step 1.3. This is the annihilator thickening, not the reduced support; it retains any nilpotents in ${\mathcal{O}}_X\mathrm{/}\mathcal{I}$. Since $\mathcal{I}$ annihilates $\mathcal{Q}$, the ${\mathcal{O}}_X$-action on $\mathcal{Q}$ factors through ${\mathcal{O}}_S$. On $S\cap U=\mathrm{Spec}(A\mathrm{/}{J}_U)$ define $\mathcal{G}$ by the same module $M$, now regarded as an $A\mathrm{/}{J}_U$-module. The restriction maps inherited from $\mathcal{Q}$ are ${\mathcal{O}}_S$-linear because $\mathcal{I}$ annihilates $\mathcal{Q}$, and their cocycle identities are inherited from those of $\mathcal{Q}$; hence they glue the local modules to a quasi-coherent ${\mathcal{O}}_S$-module $\mathcal{G}$. For every principal open $D(f)\subseteq U$, the direct image definition and the associated-sheaf section formula identify $(i_{\ast }\mathcal{G})(D(f))=M_{\bar{f}}=M_f=\mathcal{Q}(D(f))$, compatibly with restrictions; hence $i_{\ast }\mathcal{G}\cong \mathcal{Q}$.

Euler characteristic of the quotient. By step 2.2 only $H^0$ contributes, so $\chi (X,\mathcal{Q})={\sum }_{x\in S}{\delta }_x(X)$, the sum being finite by step 1.3.

Arithmetic and geometric genus. Substituting steps 1.4 and 3.1 into step 2.1 gives $\chi (X,{\mathcal{O}}_X)=\chi (X^{\mathrm{n}\mathrm{u}},{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}})-{\sum }_x{\delta }_x(X)$. By [F3] one has $1-p_a(X)=\chi (X,{\mathcal{O}}_X)$ and, since $H^0(X^{\mathrm{n}\mathrm{u}},{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}})=k$ [F3], also $\chi (X^{\mathrm{n}\mathrm{u}},{\mathcal{O}}_{X^{\mathrm{n}\mathrm{u}}})=1-g(X^{\mathrm{n}\mathrm{u}})$. Therefore $1-p_a(X)=1-g(X^{\mathrm{n}\mathrm{u}})-{\sum }_x{\delta }_x(X)$, that is, $p_a(X)=g(X^{\mathrm{n}\mathrm{u}})+{\sum }_x{\delta }_x(X)$.

Conclusion. For an integral proper finite-type curve over an algebraically closed field, the arithmetic genus exceeds the geometric genus of the normalization exactly by the total delta invariant, $p_a(X)=g(X^{\mathrm{n}\mathrm{u}})+{\sum }_x{\delta }_x(X)$, the sum finite and supported on the singular points by step 1.3; equivalently $g(X^{\mathrm{n}\mathrm{u}})=p_a(X)-{\sum }_x{\delta }_x(X)$. The Axiom of Choice is inherited from the normalization, finiteness and cohomology suppliers used in steps 1.1, 1.4 and 2.2. ∎