Weighted complete intersections and lattice points (Q1346345)

Let \(S\) be a quasismooth weighted complete intersection surface of multidegree \((d_ 1, \dots, d_{n-2})\) in a weighted projective space \(\mathbb{P} (q_ 0, \dots, q_ n)\), such that \(\text{lcm} \{q_ i\}\) divides all \(d_ j\). Using two results which \textit{F. Hirzebruch} and \textit{P. Zagier} established as corollaries of the Atiyah-Singer theorem [cf. ``The Atiyah-Singer theorem and elementary number theory'' (1974; Zbl 0288.10001)], an explicit expression for the geometric genus of \(S\) is given. By an elementary observation, this expression implies the Mordell-Pommersheim formula on the number of lattice points in the tetrahedron in \(\mathbb{Z}^ 3\) with vertices (0,0,0), \((a,0,0)\), \((0,b,0)\) and \((0,0,c)\).