Let \(C\) be a smooth curve on a smooth projective surface \(X\). Then \[ \omega_C = (\omega_X \otimes \OO_X(C))\vert_C , \qquad 2g(C) - 2 = C \cdot (C + K_X) . \] The same numerical formula computes the arithmetic genus of any curve on \(X\).
Adjunction converts genus calculations on surfaces into intersection-number calculations.
On \(\PP^2\) it gives the degree-genus formula. On \(\PP^2\) one has \(K = -3H\) and \(H^2 = 1\), so \(K^2 = 9\); for \(C\) of degree \(d\), \[ 2g-2 = C \cdot (C - 3H) = d^2 - 3d , \] giving \(g = \tfrac{1}{2}(d-1)(d-2)\). On \(\PP^1 \times \PP^1\), \(K = (-2,-2)\) and the same computation gives \(g = (a-1)(b-1)\) for a curve of type \((a,b)\).
In any dimension, for a smooth divisor \(D\) on a smooth \(X\), the sheaf-level form is \(K_D = (K_X + D)\vert_D\).
Dependencies
D-SRFINTThe intersection pairing on a surfaceuses
Backlinks
FE-CRVQUADCurves of type $(a,b)$ on a quadric surfaceusesT-SRFRRRiemann--Roch for surfacesusesFE-SRFFERMATQUARTICThe Fermat quartic is a K3 surface containing $48$ linesusesFE-SRFCUBICThe cubic surface and its $27$ linesuses