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3

$begingroup$

Let $f_i(x,y,z)$ be a homogeneous polynomial of degree $i$. Then for pencil $C_a,b=a(f_2)^3+b(f_3)^2$, we have a map $phi:mathbbCP^2setminus Bto mathbbCP^1$, where $B$ is the base locus of pencil $C_a,b$, and the map is given by $[x,y,z]mapsto [a,b]=[(f_3)^2(x,y,z):(f_2)^3(x,y,z)]$.

Let $C$ be the union $phi^−1(1,0)cup phi^−1(0,1)cupphi^−1 (1,1)$ and let $L$ be a line containing only smooth points of $C$ and transversal to $C$.

Then how to see the map: $pi_1(Lsetminus Lcap C)topi_1(mathbbCP^2setminus C)$ induced by inclusion is surjective? Moreover, how to compute these two fundamental groups?

I guess we can use Lefschetz hyperplane theorem to deduce the surjection. But can't go further .

algebraic-geometry algebraic-topology projective-space fundamental-groups

share|cite|improve this question

edited Mar 16 at 21:38

6666

asked Mar 16 at 20:56

66666666

1,375621

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I should also say that it can be very complicated to describe the fundamental group of $mathbbCP^2 setminus C$ where $C$ is an algebraic curve if $C$ is singular, the group will depend on the precise singularities that the curve (it is still not completely understood). However, if $C$ is a degree $d$ smooth curve in $mathbbCP^2$ then $pi_1(mathbbCP^2 setminus C) cong mathbbZ_d$
  $endgroup$
  – Nick L
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @NickL $f_3$ has degree $3$, and $f_3^2$ has degree $6$
  $endgroup$
  – 6666
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I see, I will delete my first comment.
  $endgroup$
  – Nick L
  2 days ago
  

  
  
  
  

add a comment | 

3

$begingroup$

Let $f_i(x,y,z)$ be a homogeneous polynomial of degree $i$. Then for pencil $C_a,b=a(f_2)^3+b(f_3)^2$, we have a map $phi:mathbbCP^2setminus Bto mathbbCP^1$, where $B$ is the base locus of pencil $C_a,b$, and the map is given by $[x,y,z]mapsto [a,b]=[(f_3)^2(x,y,z):(f_2)^3(x,y,z)]$.

Let $C$ be the union $phi^−1(1,0)cup phi^−1(0,1)cupphi^−1 (1,1)$ and let $L$ be a line containing only smooth points of $C$ and transversal to $C$.

Then how to see the map: $pi_1(Lsetminus Lcap C)topi_1(mathbbCP^2setminus C)$ induced by inclusion is surjective? Moreover, how to compute these two fundamental groups?

I guess we can use Lefschetz hyperplane theorem to deduce the surjection. But can't go further .

algebraic-geometry algebraic-topology projective-space fundamental-groups

share|cite|improve this question

edited Mar 16 at 21:38

6666

asked Mar 16 at 20:56

66666666

1,375621

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I should also say that it can be very complicated to describe the fundamental group of $mathbbCP^2 setminus C$ where $C$ is an algebraic curve if $C$ is singular, the group will depend on the precise singularities that the curve (it is still not completely understood). However, if $C$ is a degree $d$ smooth curve in $mathbbCP^2$ then $pi_1(mathbbCP^2 setminus C) cong mathbbZ_d$
  $endgroup$
  – Nick L
  2 days ago
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @NickL $f_3$ has degree $3$, and $f_3^2$ has degree $6$
  $endgroup$
  – 6666
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I see, I will delete my first comment.
  $endgroup$
  – Nick L
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

Let $f_i(x,y,z)$ be a homogeneous polynomial of degree $i$. Then for pencil $C_a,b=a(f_2)^3+b(f_3)^2$, we have a map $phi:mathbbCP^2setminus Bto mathbbCP^1$, where $B$ is the base locus of pencil $C_a,b$, and the map is given by $[x,y,z]mapsto [a,b]=[(f_3)^2(x,y,z):(f_2)^3(x,y,z)]$.

Let $C$ be the union $phi^−1(1,0)cup phi^−1(0,1)cupphi^−1 (1,1)$ and let $L$ be a line containing only smooth points of $C$ and transversal to $C$.

Then how to see the map: $pi_1(Lsetminus Lcap C)topi_1(mathbbCP^2setminus C)$ induced by inclusion is surjective? Moreover, how to compute these two fundamental groups?

I guess we can use Lefschetz hyperplane theorem to deduce the surjection. But can't go further .

algebraic-geometry algebraic-topology projective-space fundamental-groups

share|cite|improve this question

edited Mar 16 at 21:38

6666

asked Mar 16 at 20:56

66666666

1,375621

$endgroup$

share|cite|improve this question

edited Mar 16 at 21:38

6666

asked Mar 16 at 20:56

66666666

1,375621

share|cite|improve this question

edited Mar 16 at 21:38

6666

asked Mar 16 at 20:56

66666666

1,375621

asked Mar 16 at 20:56

66666666

1,375621

asked Mar 16 at 20:56

66666666

1,375621