Monoidal transforms and invariants of singularities in positive characteristic (Q2852240)

Let \(X\) be a closed subvariety of a smooth variety over a perfect field \(k\), \(m\) the maximum value of the multiplicity \(m(x)\) of \(X\) at point \(x\), and \(M(X)\) the (closed) subset of \(X\) of points \(y\) where \(m(y)=m\). A possible problem is to find a sequence of monoidal transforms \(V=V_0, \ldots, V_r\), such that if \(X_i\) is the strict transform of \(X\) to \(V_i\), the centers satisfy \(C_i \subset M(X_i)\) and the maximum multiplicity of \(X_r\) is \(<m\). If this can be done, iterating the process we resolve the singularities of \(X\).NEWLINENEWLINEThe problem has affirmative solution in characteristic zero and remains open, for \(\dim(X) > 3\), in positive characteristic. The present paper is a contribution to this open problem when \(X\) is a hypersurface in a smooth variety \(V\). As in other articles of the authors, the question is rephrased in terms of \textit{Rees algebras} (certain graded subagebras of the polynonial algebra \({\mathcal O}_V[T]\)).NEWLINENEWLINEIn previous work the authors (and Ana Bravo) have shown that, in any characteristic, it is possible to ``improve'' the singularities of a Rees algebra \(\mathcal G\) (over a regular variety \(V\)). More precisely, let \(e(\mathcal G)\) denote the minimum value of \({\tau}_{\mathcal G,x}\), for \(x\) a singular point of \( \mathcal G\), (\(\tau\) is a version of a numerical invariant introduced by Hironaka). By descending induction on \(e(\mathcal G)\) it is possible to associate to \(\mathcal G\), locally in the etale topology: (a) sequences NEWLINE\[NEWLINE(1) \qquad V=V_0 \leftarrow \cdots \leftarrow V_s \, , NEWLINE\]NEWLINE NEWLINE\[NEWLINE(2) \qquad V'_0 \leftarrow \cdots \leftarrow V'_s \, ,NEWLINE\]NEWLINE of monoidal transforms with regular centers, (c) transversal smooth projections \(\beta _i:V_i \to V'_i\), \(\dim (V'_i)=\dim(V_i)-e\), for all \(i\), and (d) Rees algebras \({\mathcal R}_{i}\), such that \({\mathcal R}_s\) is monomial. The algebra \(\mathcal{R}_i\) is the \textit{elimination algebra of} \({\mathcal G}_i\) (the transform of \({\mathcal G}\) to \(V_i\)) with respect to \(\beta _i\), a concept thoroughly studied by Villamayor. Monomial algebras correspond to \textit{monomial ideals}, a particularly simple type of invertible sheaf, which can be resolved pretty easily essentially in a combinatorial way. In this case, one says that \(\mathcal G _s\) is in the \textit{monomial situation}. If the characteristic of the base field \(k\) is zero, the mentioned result leads to an extension of the sequence (1) which resolves \(\mathcal G\), but in positive characteristic there are difficulties. Trying to overcome these, in the present paper the authors introduce a concept of ``strong monomial situation''. If in (1) \(\mathcal G _s\) is in the strongly monomial situation, even in positive characteristic it is possible to extend the sequence (1) so that a resolution of \(\mathcal G\) is achieved. The authors give a numerical criterion to decide whether the strong monomial situation has been reached.NEWLINENEWLINEBoth the notion of strong monomial situation and the mentioned criterion require some new auxiliary concepts, introduced and developed in this paper. For instance: (i) the notion of \textit{\(p\)-presentation} of a Rees algebra \(\mathcal G\) over \(V\) of dimension \(d\), relative to a transversal projection \(\beta : V \to V'\), \(\dim V' = d-1\), which is a way to describe \(\mathcal G\) (locally, in the etale topology) in terms of the elimination algebra \( \mathcal R\) of \(\mathcal G\) and a monic polynomial, whose degree is a power of \(p\), with coefficients defined on \(V'\); (ii) the notion of \textit{slope} of \(\mathcal G\) relative to a p-presentation; (iii) the notion of H-ord at a point of \(V\), defined in terms of slopes of suitable general presentations. The authors sometimes work under the assumption that \(e(\mathcal G)=1\), and announce that the general situation will be discussed in future papers.NEWLINENEWLINEIf \(\dim X =2\), in other articles they showed that, even in positive characteristic, with the methods just reviewed a resolution of singularities is obtained.