Let $X\subset\mathbb{P}^n$ be a projective variety and let $Y\subset X$ be the singular locus of $X$. Assume that $Y$ is smooth. I would like to know if the following are equivalent:
- $X$ has an ordinary singularity along $Y$,
- for any $x\in Y$ the projective tangent cone to $X$ at $x$ is smooth,
- Let $\pi:Z\rightarrow \mathbb{P}^n$ be the blow-up of $Y$ with exceptional divisor $E$. Then the strict transform of $X$ in $Z$ is smooth and intersects $E$ transversally.
I am also a little bit confused about the definition of ordinary singularity. For instance, consider the hypersurface $X$ in $\mathbb{P}^4$ given by: $$x_0x_1x_2x_3+x_0x_1x_2x_4+x_0x_1x_3x_4+x_0x_2x_3x_4+x_1x_2x_3x_4 = 0.$$ In some references I found that $X$ has ordinary triple points in the fundamental points of $\mathbb{P}^4$. On the other hand in such points the projective tangent cone is an irreducible cubic with four singular points. In other places they gave the smoothness of the projective tangent cone as definition of ordinary singularity.