Theorem
Let \(X\) be a smooth curve of genus \(g\) and \(K_X\) be its canonical sheaf. Then for any line bundle \(L\) on \(X\) of degree \(d\), \[h^0(X,L) - h^0(X, K_X - L) = d - g + 1\] or by Serre duality \[h^0(X,L) - h^1(X, L) = d - g + 1.\]
Theorem
Let \(X\) be a smooth curve of genus \(g\) and \(K_X\) be its canonical sheaf. Then for any line bundle \(L\) on \(X\) of degree \(d\), \[h^0(X,L) - h^0(X, K_X - L) = d - g + 1\] or by Serre duality \[h^0(X,L) - h^1(X, L) = d - g + 1.\]