The nonexistence of negative weight derivations on positive dimensional isolated singularities: generalized Wahl conjecture. (English) Zbl 1451.13071

Let \(R = \bigoplus_{\nu\geq 0} R_\nu\) be a graded finitely generated algebra over \(R_0=\mathbb C\). Then there is a presentation \(R = P/I\), where \(P\) is a graded polynomial ring over \(\mathbb C\) in \(n\) variables with positive weights \(w_1,\dots, w_n\) and \(I\) is an ideal generated by a sequence of quasihomogeneous polynomials \(f_1, \cdots, f_m\). In 1983, J. M. Wahl conjectured that for any normal graded ring \(R\) having an isolated singularity there is a normal graded ring \(\tilde R\) with isomorphic completion, \(R^\wedge \cong \tilde R^\wedge\), such that the module of \(\mathbb C\)-derivations \(\mathrm{Der}(\tilde R)\) has no elements of negative weight (see [J. M. Wahl, Proc. Symp. Pure Math. 40, 613–624 (1983; Zbl 0534.14001)]). On the other hand, in 1982 the reviewer has proved the following statement (in the local analytic setting): if the ideal \(I\) is generated by a regular sequence then the quotient \(\mathrm{Der}(R)/\mathrm{Der^o}(R)\), where \(\mathrm{Der^o}(R)\) is the module of trivial \(\mathbb C\)-derivations, has no elements of negative weight in the induced grading because the quotient is generated by the Euler derivation. Moreover, in this case there are many examples of graded algebras with trivial derivations of negative weight (see [A. G. Aleksandrov, Math. USSR, Izv. 26, 437–477 (1986; Zbl 0647.14027); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 3, 467–510 (1985)]). Similar results one can easily obtain for any Gorenstein 1-dimensional graded analytic algebra \(R\) using an explicit description of the above quotient (see [E. Kunz and R. Waldi, Math. Z. 187, 105–123 (1984; Zbl 0543.14014)]). In the paper under review the authors assert that the module \(\mathrm{Der}(R)\) has no elements of negative weight for any graded \(\mathbb C\)-algebra \(R\) satisfying the following two conditions: it determines an isolated singularity of positive dimension and its defining ideal \(I\) can be generated by a sequence of polynomials whose weighted degrees are high enough. In addition, the corresponding bound is expressed explicitly in terms of weights of variables of the ambient ring. It should be noted that the proof presented is mainly based on a series of purely combinatorial arguments and involves a lot of tedious calculations.


13N05 Modules of differentials
14B05 Singularities in algebraic geometry
32S05 Local complex singularities
Full Text: DOI Euclid


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