Let \(k\) be a field.
(Degree of a closed point)
Let \(X\) be a scheme locally of finite type over \(k\) and \(p \in X\) a closed point. The degree of \(p\) is \(\deg p = [\kappa(p) : k]\), which is finite by the Nullstellensatz.
(Degree of a finite morphism)
Let \(\pi \colon X \to Y\) be a finite morphism and \(y \in Y\). The degree of \(\pi\) at \(y\) is \(\dim_{\kappa(y)} (\pi_* \OO_X \otimes_{\OO_Y} \kappa(y))\). Since \(\pi\) is affine, \(\pi^{-1}(y) = \Spec (\pi_* \OO_X \otimes \kappa(y))\), so this is the dimension of the ring of functions on the fibre. If \(X\) is a curve without embedded points, \(Y\) is a regular curve and \(\pi\) is finite and surjective, then \(\pi_* \OO_X\) is locally free of finite rank, and its rank, the degree at every point, is the degree \(\deg \pi\).
(Degree of a line bundle and of a coherent sheaf)
Let \(C\) be a projective curve over \(k\). The degree of an invertible sheaf \(\mathcal{L}\) on \(C\) is \(\deg_C \mathcal{L} = \chi(C, \mathcal{L}) - \chi(C, \OO_C)\). If \(C\) is integral, the degree of a coherent sheaf \(\mathcal{F}\) on \(C\) is \(\deg \mathcal{F} = \chi(C, \mathcal{F}) - (\operatorname{rank} \mathcal{F}) \chi(C, \OO_C)\), where \(\operatorname{rank} \mathcal{F}\) is the dimension of the stalk of \(\mathcal{F}\) at the generic point over the function field. For rank \(1\) locally free \(\mathcal{F}\) the two definitions agree.
Let \(C\) be a regular projective integral curve over \(k\).
For a divisor \(D = \sum_i n_i p_i\), \(\deg \OO_C(D) = \deg D = \sum_i n_i \deg p_i\) (PR-Y5S7V); so if \(s\) is a nonzero rational section of \(\mathcal{L}\), then \(\deg \mathcal{L} = \deg \div s\).
For a locally free sheaf \(\mathcal{E}\) of rank \(r\) and an invertible sheaf \(\mathcal{L}\), \(\chi(\mathcal{E} \otimes \mathcal{L}) = \chi(\mathcal{E}) + r \deg \mathcal{L}\).
If \(\pi \colon C' \to C\) is a finite surjective morphism from a projective integral curve, then \(\deg_{C'} \pi^* \mathcal{L} = (\deg \pi)(\deg_C \mathcal{L})\) [@Har10a, Proposition II.6.9].
Dependencies
D-COHEULERThe Euler characteristic and the Hilbert polynomialuses