This paper of Zariski contains this statement: If $C$ is a curve in $\mathbb{CP}^2$, and $L$ is a generic line, then the injection $L\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces an isomorphism of fundamental groups.

1. Edited version: Let $C$ be a fixed curve (not necessarily smooth) in $\mathbb{CP}^2$. Does there exist a characterization of curves $D$ such that the injection $D\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces a surjection of fundamental groups? In other words, how to check if every loop in $\mathbb{CP}^2\setminus C$ can be deformed into a loop in $D\setminus C$?

2. More precisely, if $V$ is a base point free linear family of curves, then is it true that the answer to question 1 will be true for generic elements in $V$? This question has a positive answer if $V$ is a pencil of lines (by the statement written in the first line) - so the question is what happens in other cases.

3. How about Questions 1 and 2 with $\mathbb{CP}^2$ replaced by $\mathbb{C}^2$?

I just got interested into this sort of questions, and am a bit overwhelmed with the vast literature on topology of complements of hypersurfaces. Any introductory reference would be greatly appreciated as well.

This paper of Zariski contains this statement: If $C$ is a curve in $\mathbb{CP}^2$, and $L$ is a generic line, then the injection $L\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces an isomorphism a surjection of fundamental groups.

1. Edited version: Let $C$ be a fixed curve (not necessarily smooth) in $\mathbb{CP}^2$. Does there exist a characterization of curves $D$ such that the injection $D\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces a surjection of fundamental groups? In other words, how to check if every loop in $\mathbb{CP}^2\setminus C$ can be deformed into a loop in $D\setminus C$?

2. More precisely, if $V$ is a base point free linear family of curves, then is it true that the answer to question 1 will be true for generic elements in $V$? This question has a positive answer if $V$ is a pencil of lines (by the statement written in the first line) - so the question is what happens in other cases.

3. How about Questions 1 and 2 with $\mathbb{CP}^2$ replaced by $\mathbb{C}^2$?

I just got interested into this sort of questions, and am a bit overwhelmed with the vast literature on topology of complements of hypersurfaces. Any introductory reference would be greatly appreciated as well.

This paper of Zariski contains this statement: If $C$ is a curve in $\mathbb{CP}^2$, and $L$ is a generic line, then the injection $L\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces an isomorphism of fundamental groups.

1. Does there exist a characterization of curves $D$ such that the injection $D\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces a surjection of fundamental groups?

2. More precisely, if $V$ is a base point free linear family of curves, then is it true that the answer to question 1 will be true for generic elements in $V$?

3. How about Questions 1 and 2 with $\mathbb{CP}^2$ replaced by $\mathbb{C}^2$?

I just got interested into this sort of questions, and am a bit overwhelmed with the vast literature on topology of complements of hypersurfaces. Any introductory reference would be greatly appreciated as well.

This paper of Zariski contains this statement: If $C$ is a curve in $\mathbb{CP}^2$, and $L$ is a generic line, then the injection $L\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces an isomorphism of fundamental groups.

1. Edited version: Let $C$ be a fixed curve (not necessarily smooth) in $\mathbb{CP}^2$. Does there exist a characterization of curves $D$ such that the injection $D\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces a surjection of fundamental groups? In other words, how to check if every loop in $\mathbb{CP}^2\setminus C$ can be deformed into a loop in $D\setminus C$?

2. More precisely, if $V$ is a base point free linear family of curves, then is it true that the answer to question 1 will be true for generic elements in $V$? This question has a positive answer if $V$ is a pencil of lines (by the statement written in the first line) - so the question is what happens in other cases.

3. How about Questions 1 and 2 with $\mathbb{CP}^2$ replaced by $\mathbb{C}^2$?

I just got interested into this sort of questions, and am a bit overwhelmed with the vast literature on topology of complements of hypersurfaces. Any introductory reference would be greatly appreciated as well.