(Geometric) Genus of a curve equals to -1 in SageMath
 
  
 
  
 
 
 
 
 
 
    
        
        asked 5 years ago  
 
 azerbajdzan gravatar image  
 
 
 
 
    
        
        updated 5 years ago  
 
 
 
 
 
 
What does it mean when Sage computes
  genus of a projective plane curve to
  be -1?
 
 
Input:
 
x,y,z = QQ['x,y,z'].gens()
C = Curve(x^4 + 10*x^2*y*z + 5*y^2*z^2)
C.genus()
 
Output:
 
-1
 
Preview: (hide)
 
 
 
 
          
 
Comments
Not a smooth curve.
sage: C.is_smooth()
False
FrédéricC gravatar imageFrédéricC (
    
        5 years ago
    )
1
C is not geometrically irreducible: over Q(√5) it splits into two components. So, the Euler characteristic of the desingularization ends up being χ=−4. Using the formula χ=2g−2, you end up with genus −1. I'm not saying this is a mathematically correct answer; just that this is probably how the answer is discovered: basically by computing the degree of the canonical bundle.
nbruin gravatar imagenbruin (
    
        5 years ago
    )
add a comment