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The following Theorem by Nori (Proposition 3.27 of this paper) is in the spirit of what you are looking for.

Theorem. Let $D$ and $E$ be curves on a smooth projective surface $X$. Assume that

1. the only singularities of $D$ are nodes;
2. $D$ and $E$ intersect transversally;
3. every irreducible component $C$ of $D$ satisfies $C^2 > 2 r(C)$ where $r(C)$ is the number of singularities of $C$.

Then the kernel of the natural morphism $$ \pi_1(X-D-E) > \longrightarrow \pi_1(X-E) $$ is abelian.

In particular if you take an irreducible plane curve $C$ having a triple point with distinct tangents as its only singularity then the fundamental group of the complement of its strict transform is abelian (apply Theorem above to $D = \hat C$ and $ E = \emptyset$ ). As Dmitri pointed out, if you drop irreducibility this is no longer true.

You might want to take a look at this survey. There you will find Nori's Theorem in Section 2.3.

Further comments added later:

1. If $C \subset \mathbb P^2$ is an irreducible curve with only one singularity having smooth branches intersecting pairwise transversely then the complement of $C$ in $\mathbb P^2$ and well as the complement of its strict transform in the blow-up of $\mathbb P^2$ are abelian ($D = \hat C$, $E =$ exceptional divisor in the first case; and $D = \hat C$, $E= \emptyset$ in the second).
2. If you have a reduced connected curve $C = E_1 + \ldots + E_k$ with rational irreducible components, the intersection matrix $(E_i\cdot E_j)$ is negative definite, and the dual graph is a tree then the fundamental group of the complement of a neighborhood of $C$ has been determined by Mumford in this paper.

The following Theorem by Nori (Proposition 3.27 of this paper) is in the spirit of what you are looking for.

Theorem. Let $D$ and $E$ be curves on a smooth projective surface $X$. Assume that

1. the only singularities of $D$ are nodes;
2. $D$ and $E$ intersect transversally;
3. every irreducible component $C$ of $D$ satisfies $C^2 > 2 r(C)$ where $r(C)$ is the number of singularities of $C$.

Then the kernel of the natural morphism $$ \pi_1(X-D-E) > \longrightarrow \pi_1(X-E) $$ is abelian.

In particular if you take an irreducible plane curve $C$ having a triple point with distinct tangents as its only singularity then the fundamental group of the complement of its strict transform is abelian (apply Theorem above to $D = \hat C$ and $ E = \emptyset$ ). As Dmitri pointed out, if you drop irreducibility this is no longer true.

You might want to take a look at this survey. There you will find Nori's Theorem in Section 2.3.

Further comments added later:

1. If $C \subset \mathbb P^2$ is an irreducible curve with only one singularity having smooth branches intersecting pairwise transversely then the complement of $C$ in $\mathbb P^2$ and well as the complement of its strict transform in the blow-up of $\mathbb P^2$ are abelian ($D = \hat C$, $E =$ exceptional divisor in the first case; and $D = \hat C$, $E= \emptyset$ in the second).
2. If you have a reduced connected curve $C = E_1 + \ldots + E_k$ with rational irreducible components, the intersection matrix $(E_i\cdot E_j)$ is negative definite, and the dual graph is a tree then the fundamental group of the complement of a neighborhood of $C$ has been determined by Mumford in this paper.

The following Theorem by Nori (Proposition 3.27 of this paper) is very much in the spirit of what you are looking for.

Theorem. Let $D$ and $E$ be curves on a smooth projective surface $X$. Assume that

1. the only singularities of $D$ are nodes;
2. $D$ and $E$ intersect transversally;
3. every irreducible component $C$ of $D$ satisfies $C^2 > 2 r(C)$ where $r(C)$ is the number of singularities of $C$.

Then the kernel of the natural morphism $$ \pi_1(X-D-E) > \longrightarrow \pi_1(X-E) $$ is abelian.

In particular if you take an irreducible plane curve $C$ having a triple point with distinct tangents as its only singularity then the fundamental group of the complement of its strict transform is abelian (apply Theorem above to $D = \hat C$ and $ E = \emptyset$ ). As Dmitri pointed out, if you drop irreducibility this is no longer true.

You might want also to take a look at this survey. There you will find Nori's Theorem in Section 2.3.

The following Theorem by Nori (Proposition 3.27 of this paper) is in the spirit of what you are looking for.

Theorem. Let $D$ and $E$ be curves on a smooth projective surface $X$. Assume that

1. the only singularities of $D$ are nodes;
2. $D$ and $E$ intersect transversally;
3. every irreducible component $C$ of $D$ satisfies $C^2 > 2 r(C)$ where $r(C)$ is the number of singularities of $C$.

Then the kernel of the natural morphism $$ \pi_1(X-D-E) > \longrightarrow \pi_1(X-E) $$ is abelian.

In particular if you take an irreducible plane curve $C$ having a triple point with distinct tangents as its only singularity then the fundamental group of the complement of its strict transform is abelian (apply Theorem above to $D = \hat C$ and $ E = \emptyset$ ). As Dmitri pointed out, if you drop irreducibility this is no longer true.

You might want to take a look at this survey. There you will find Nori's Theorem in Section 2.3.

Further comments added later:

1. If $C \subset \mathbb P^2$ is an irreducible curve with only one singularity having smooth branches intersecting pairwise transversely then the complement of $C$ in $\mathbb P^2$ and well as the complement of its strict transform in the blow-up of $\mathbb P^2$ are abelian ($D = \hat C$, $E =$ exceptional divisor in the first case; and $D = \hat C$, $E= \emptyset$ in the second).
2. If you have a reduced connected curve $C = E_1 + \ldots + E_k$ with rational irreducible components, the intersection matrix $(E_i\cdot E_j)$ is negative definite, and the dual graph is a tree then the fundamental group of the complement of a neighborhood of $C$ has been determined by Mumford in this paper.

The following Theorem by Nori (Proposition 3.27) is very much in the spirit of what you are looking for.

Theorem. Let $D$ and $E$ be curves on a smooth projective surface $X$. Assume that

1. the only singularities of $D$ are nodes;
2. $D$ and $E$ intersect transversally;
3. every irreducible component $C$ of $D$ satisfies $C^2 > 2 r(C)$ where $r(C)$ is the number of singularities of $C$.

Then the kernel of the natural morphism $$ \pi_1(X-D-E) > \longrightarrow \pi_1(X-E) $$ is abelian.

In particular if you take an irreducible plane curve having a triple point with distinct tangents as its only singularity then the fundamental group of the complement of its strict transform is abelian. As Dmitri pointed out, if you drop irreducibility this is no longer true.

You might want also to take a look at this survey. There you will find Nori's Theorem in Section 2.3.

The following Theorem by Nori (Proposition 3.27 of this paper) is very much in the spirit of what you are looking for.

Theorem. Let $D$ and $E$ be curves on a smooth projective surface $X$. Assume that

1. the only singularities of $D$ are nodes;
2. $D$ and $E$ intersect transversally;
3. every irreducible component $C$ of $D$ satisfies $C^2 > 2 r(C)$ where $r(C)$ is the number of singularities of $C$.

Then the kernel of the natural morphism $$ \pi_1(X-D-E) > \longrightarrow \pi_1(X-E) $$ is abelian.

In particular if you take an irreducible plane curve $C$ having a triple point with distinct tangents as its only singularity then the fundamental group of the complement of its strict transform is abelian (apply Theorem above to $D = \hat C$ and $ E = \emptyset$ ). As Dmitri pointed out, if you drop irreducibility this is no longer true.

You might want also to take a look at this survey. There you will find Nori's Theorem in Section 2.3.