linear equivalence of divisors
Two divisors can look different point-by-point yet be 'the same' for every geometric purpose. The reason: moving the zeros and poles of a meromorphic function around does not change anything intrinsic, because you can always multiply by that function. Linear equivalence is the precise way of saying two divisors differ only by such a harmless move — by the divisor of some meromorphic function — and so should be treated as one class.
Precisely, two divisors D and D' on a compact Riemann surface are linearly equivalent, written D ~ D', if their difference is a principal divisor: D - D' = div(f) for some nonzero meromorphic function f. This is an equivalence relation (reflexive via f = 1, symmetric via 1/f, transitive via products). Because every principal divisor has degree 0 on a compact surface, linearly equivalent divisors always have the SAME degree — degree is a linear-equivalence invariant. The set of effective divisors linearly equivalent to a fixed D is the complete linear system |D|, and it is naturally a projective space P(L(D)): each nonzero function in L(D) gives an effective divisor D + div(f), and proportional functions give the same divisor.
Why it matters: linear equivalence is the relation that organizes divisors into a usable group. The quotient of all divisors by principal divisors is the Picard group Pic(X), and within it the degree-d classes form a torsor under the degree-0 part Pic^0(X), which is the Jacobian. Riemann-Roch depends only on the linear-equivalence class. The canonical divisor is defined only up to linear equivalence. A genuine subtlety: on the sphere (g = 0) two divisors are linearly equivalent IFF they have equal degree — degree is a complete invariant. But for g >= 1 that fails: equal degree is necessary but not sufficient, and the extra obstruction is measured by the Abel-Jacobi map landing in the Jacobian. So 'same degree' and 'linearly equivalent' coincide only in genus 0.
On the sphere, [0] ~ [infinity] because f(z) = z has div(f) = [0] - [infinity]. So any single point is linearly equivalent to any other, and every degree-1 divisor on the sphere is linearly equivalent. On a torus, however, [p] ~ [q] holds only when p = q: distinct points are NEVER linearly equivalent, which is exactly why the torus has a nontrivial Jacobian.
On the sphere all degree-d divisors are equivalent; on a torus distinct points already differ — the gap is the Jacobian.
Equal degree forces linear equivalence ONLY in genus 0. For g >= 1, two divisors of the same degree can be linearly inequivalent; the obstruction is precisely their difference's image under the Abel-Jacobi map into Pic^0 = the Jacobian.