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It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, containing a finite number of pinch points and a finite number of triple points, which are triple also for $S'$. The proof is obtained by embedding $S$ in $\mathbb{P}^5$ and by taking a projection

$\pi_{L} \colon S \to \mathbb{P}^3$,

where $L \subset \mathbb{P}^5$ is a general line. This is the method originally used by M. Noether in order to prove his famous formula

$\chi(\mathcal{O}_S)=\frac{1}{12}(K_S^2+c_2(S))$,

see Griffiths-Harris "Principles of Algebraic Geometry", p. 600.

My question is now the following:

is it also true that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only isolated singularities? And if not, is there any counterexample?

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, containing a finite number of pinch points and a finite number of triple points, which are triple also for $S'$. The proof is obtained by embedding $S$ in $\mathbb{P}^5$ and by taking a projection $$\pi_{L} \colon S \to \mathbb{P}^3,$$ where $L \subset \mathbb{P}^5$ is a general line. This is the method originally used by M. Noether in order to prove his famous formula $$\chi(\mathcal{O}_S)=\frac{1}{12}(K_S^2+c_2(S)),$$ see Griffiths-Harris Principles of Algebraic Geometry, p. 600.

Question. Is it also true that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only isolated singularities? And if not, what is a counterexample?

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only $ordinary$ $singularities$, i.e. a curve $C$ of double points, containing a finite number of pinch points and a finite number of triple points, which are triple also for $S'$. The proof is obtained by embedding $S$ in $\mathbb{P}^5$ and by taking a projection

$\pi_{L} \colon S \to \mathbb{P}^3$,

where $L \subset \mathbb{P}^5$ is a general line. This is the method originally used by M. Noether in order to prove his famous formula

$\chi(\mathcal{O}_S)=\frac{1}{12}(K_S^2+c_2(S))$,

see Griffiths-Harris "Principles of Algebraic Geometry", p. 600.

My question is now the following:

is it also true that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only $isolated$ singularities? And if not, is there any counterexample?

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, containing a finite number of pinch points and a finite number of triple points, which are triple also for $S'$. The proof is obtained by embedding $S$ in $\mathbb{P}^5$ and by taking a projection

$\pi_{L} \colon S \to \mathbb{P}^3$,

where $L \subset \mathbb{P}^5$ is a general line. This is the method originally used by M. Noether in order to prove his famous formula

$\chi(\mathcal{O}_S)=\frac{1}{12}(K_S^2+c_2(S))$,

see Griffiths-Harris "Principles of Algebraic Geometry", p. 600.

My question is now the following:

is it also true that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only isolated singularities? And if not, is there any counterexample?

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only $ordinary$ $singularities$, i.e. a curve $C$ of double points, containing a finite number of pinch points and a finite number of triple points, which are triple also for $S'$. The proof is obtained by embedding $S$ in $\mathbb{P}^5$ and by taking a projection

$\pi_{L} \colon S \to \mathbb{P}^3$,

where $L \subset \mathbb{P}^5$ is a general line. This is the method originally used by M. Noether in order to prove his famous formula

$\chi(\mathcal{O}_S)=\frac{1}{12}(K_S^2+c_2(S))$,

see Griffiths-Harris "Principles of Algebraic Geometry", p. 600.

My question is now the following:

is it also true that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only $isolated$ singularities? And if not, is there any counterexample?

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only $ordinary$ $singularities$, i.e. a curve $C$ of double points, containing a finite number of pinch points and a finite number of triple points, which are triple also for $S'$. The proof is obtained by embedding $S$ in $\mathbb{P}^5$ and by taking a projection

$\pi_{L} \colon S \to \mathbb{P}^3$,

where $L \subset \mathbb{P}^5$ is a general line. This is the method originally used by M. Noether in order to prove his famous formula

$\chi(\mathcal{O}_S)=\frac{1}{12}(K_S^2+c_2(S))$,

see Griffiths-Harris "Principles of Algebraic Geometry", p. 600.

My question is now the following:

is it also true that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only $isolated$ singularities? And if not, is there any counterexample?