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Given an algebraically closed number field $k$, we consider the projective space $\mathbb P_{n+1}$, the hyperplane $H=\{X_{n+1}=0\}\simeq\mathbb P_n$ and a hypersurface $X$ defined by a polynomial $f(X_0,\ldots,X_{n+1})$. We define on $X$ the projection $\phi$ from the point $(0:\cdots :0:1)$ to $H$ which maps $(x_0:\cdots :x_n:x_{n+1})\in X$ to $(x_0:\cdots :x_n)\in H$.

My questions are: (1) how is defined the branch locus of this map? (2) how can I compute it?

My answers would be: (1) the set of points of $H$ whose preimage in $X$ is made by less than deg $\phi$ points. Am I too optimistic? (2) I look at $f$ as a polynomial in a single indeterminate $X_{n+1}$ and I compute its discriminant $\Delta$. Then the branch locus would be the union of the zero loci of $\Delta$ and of the leading coefficient, which are both polynomials in $k[X_0,\ldots,X_n]$.

Another doubt... what happens if $(0:\cdots :0:1)\in X$? Can I just define the projection as "taking the first $n+1$ coordinates"?

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