for $k=\dim(Y)$.

One can check that conditions (i) and (ii) above are satisfied if either (a) or (b) is satisfied.

The main result of this paper is the following:

\[thm:risultato\] Let $X\subset{\mathbb{P}}^n$ be a quasi-projective variety with singular locus $\Sigma_X$, $H\subset{\mathbb{P}}^n$ be a hyperplane and $Y:=X\cap H$ be the corresponding hyperplane section with singular locus $\Sigma_Y$. Denote by $\tilde{X}$ and $\tilde{Y}$ resolutions of $X$ and $Y$, and by $D_X$ and $D_Y$ the corresponding exceptional divisors.

Assume that:

(i) There exist $m$-cubical hyperresolutions $H(Y)_{{\square}}$ and $H(X)_{{\square}}$ of the resolution squares $S(Y)$ and $S(X)$ associated to $Y$ and $X$ and a closed immersion $H(Y)_{{\square}}\hookrightarrow H(X)_{{\square}}$.

(ii) There exists an $m_1$-cubical hyperresolution ${D_Y}_{{\square}}$ (resp. ${D_X}_{{\square}}$) of $D_Y$ (resp. $D_X$) and an $m_2$-cubical hyperresolution ${\Sigma_Y}_{{\square}}$ (resp. ${\Sigma_X}_{{\square}}$) of $\Sigma_Y$ (resp. $\Sigma_X$) such that:

    (I) For all $I\in{\square}^*_{m_1}$ there exists a closed immersion ${D_Y}_I\hookrightarrow {D_X}_I$ which restricts to a codimension one closed immersion on each irreducible component of ${D_Y}_I$.

    (II) There exists $c\in\{0,1\}$ such that for all $I\in{\square}^*_{m_2}$ there exists a closed immersion ${\Sigma_Y}_I\hookrightarrow {\Sigma_X}_I$ which restricts to a codimension $c$ closed immersion on each irreducible component of ${\Sigma_Y}_I$.

We have the following:

1.  If $c=1$ then there exist Gysin morphisms $H^k(Y)\rightarrow H^{k+2}(X)$ which are 