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Let $X$ be a $n$-dimensional complex projective algebraic variety, let us suppose that $X$ has only isolated singularities.

Edit: Let us say that an ordinary $m$-ple singular point is an isolated singular point such that the tangent cone to this point is the cone over a $n-1$-dimensional smooth hypersurface $H\subset \mathbb{C}P^n$ of degree $m$.

If $X$ has only such ordinary $m$-ple points what do we know about the geometry and topology of $X$ and of its resolutions?:

1) What about the Mixed Hodge structure on the singular cohomology of $X$ and of its resolutions?

2) When do small resolutions exist ($IH$-small)? If $X$ is a threefold and has only nodes then small resolutions exist, what about other multiplicities and dimensions?

3) Do there exist some relations between the number of ordinary multiple points, their multiplicities and some geometric or topological invariants of $X$?

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    $\begingroup$ You might be interested in my recent preprint On factoriality of threefolds with isolated singularities (joint work with A. Rapagnetta and P. Sabatino), where factoriality of threefold hypersurfaces with only ordinary multiple points is investigated. See arxiv.org/abs/1305.4371 $\endgroup$
    – Francesco Polizzi
    Commented Dec 6, 2013 at 21:43
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    $\begingroup$ Your question seems to be immensely wide. Maybe asking something more focused will be more successful? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Dec 8, 2013 at 18:17

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