The intersection pairing

• $\operatorname{Pic}(X^{\mathrm{al}})$ with the intersection pairing $(D,D') \mapsto D \cdot D'$.

• Even symmetric bilinear form: $D \cdot D = 2g(D) - 2$ for every curve $D$.
• $\operatorname{disc}\operatorname{Pic}(X^{\mathrm{al}}):=\det(D_i\cdot D_j)$ in a $\Z$-basis.
• Signature $(1,\rho(X^{\mathrm{al}})-1)$: Hodge index theorem.
• $\operatorname{Gal}(k^{\mathrm{al}}/k)$ permutes curves, respects linear equivalence and preserves the pairing.