Over fields of characteristic zero, resolution of singularities is achieved by means of an inductive argument, which is sustained on the existence of the so called hypersurfaces of maximal contact. We report here on an alternative approach which replaces hypersurfaces of maximal contact by generic projections. Projections can be defined in arbitrary characteristic, and this approach has led to new invariants when applied to the open problem of resolution of singularities over arbitrary fields. We show here how projections lead to a form of elimination of one variable using invariants that, to some extent, generalize the notion of discriminant.

This exposition draws special attention on this form of elimination, on its motivation, and its use as an alternative approach to inductive arguments in resolution of singularities. Using techniques of projections and elimination one can also recover some well known results. We illustrate this by showing that the Hilbert-Samuel stratum of a d-dimensional non-smooth variety can be described with equations involving at most d variables.

In addition this alternative approach, when applied over fields of characteristic zero, provides a conceptual simplification of the theorem of resolution of singularities as it trivializes the globalization of local invariants.

The reader will find in this article a collection of problems, questions and exercises related to the singularities of algebraic and analytic varieties. Many of them are inspired by the work and mathematical conception of Hironaka: they are concrete, involve basic ideas and techniques from geometry and algebra, and they can immediately be attacked from scratch. Some problems rely on or use results proven by Hironaka. Simple and double asterisques indicate the more difficult problems; they may potentially serve as a topic of future research.

The aim of this note is to use the concept of algebra husks to prove an analog of the flattening decomposition theorem for simultaneous normalizations. Other applications improve earlier flatness criteria.

A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially of finite type over a noetherian scheme $S$. The theory takes values in the category of symmetric graded modules over the graded-commutative ring $\oplus_i \mathrm{H}^i(S,\mathcal{O}_S)$. In degree $i$, the cohomology and homology $\mathrm{H}^0(S,\mathcal{O}_S)$-modules thereby associated to such an $x: X \to S$, with Hochschild complex $\mathcal{H}_x$, are $\mathrm{Ext}^i_{\mathcal{O}_X} (\mathcal{H}_x,\mathcal{H}_x)$ and $\mathrm{Ext}^{−i}_{\mathcal{O}_X} (\mathcal{H}_x, x^!\mathcal{O}_S) (i \in \mathbb{Z})$. This lays the foundation for a sequel that will treat orientations in bivariant Hochschild theory through canonical relative fundamental class maps, unifying and generalizing previously known manifestations, via differential forms, of such maps.