The genus in its several senses

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.

What is the genus of a curve?

Define the geometric genus.

What might the geometric genus of a singular curve be?

Does the genus depend on the embedding?

Show that a nonsingular projective curve has $p_a = p_g = h^1(\OO_X)$.

Dependencies

T-MWDVLRiemann--Roch and Serre duality on a curveuses
D-L6ERWThe Hilbert polynomial and its coefficientsuses

Backlinks

T-CRVEMBP3Every curve embeds in $\PP^3$ and projects to a nodal plane curveuses
D-CRVPLSINGNode, cusp, and tacnode; the delta invariant and the genus formulauses
T-BIRINVPlurigenera and irregularity are birational invariantsuses

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