Classification of 2-dimensional graded normal hypersurfaces with $a(R)\le 6$

Let k be a field of any characteristic and R = k[x,y,z]/(f) be a graded normal hypersurface. We call (a,b,c; h) = deg(x,y,z;f) the type of R with gcd(a,b,c)=1. Then the a-invariant a(R) is given by h - (a+b+c). The classification of such R (or f) was made by many authors (Arnold, Saito, Wagreich, ...). Here we classify the possible types of R for a fixed a(R) with $- 1 \le a(R) \le 6$ by commutative ring theoretic method using the Dolgachev-Pinkham-Demazure construction of normal graded rings. We also show that if we fix $a(R) \ge 0$, then the number of possible types of R is finite.

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