7CCMMS16T -- Algebraic Curves

• Office: S5.29
  
• Office hours: Fri 14:00-16:00
  
• Lecture times and place (begins on 27 Sep.): Fri 9:00-11:00 STRAND S5.20
  
• Tutorials (begins on 4 Oct.): Fri 11:00-12:00 STRAND S5.20
  
• References: These are some of the books that I will be drawing from for the lectures
  
  • Simon Donaldson, Riemann Surfaces
    
  • Ernesto Girondo, Gabino González-Diez, Introduction to Compact Riemann Surfaces and Dessins d'Enfants
    
  • Frances Kirwan, Complex algebraic curves
    

• Send me an e-mail if you are going to continue with this course.

Lecture notes

• Lecture notes

Revision lecture

Homework

• Homework I
  • Solutions

• Homework II
  • Solutions

• Homework III
  • Solutions

• Homework IV
  • Solutions

Past Exams

• Exam 2019 / hand-written solutions / pdf solutions
  

• Exam 2018 / pdf solutions
  

Computer Experiments

• Draw algebraic curves over real numbers (RR). Here is an elliptic curve:
  
  R.<x,y>=AffineSpace(RR, 2)
C = Curve([y^2 - x^3 +x], R)
C.plot()

  Try different curves. Here are some interesting ones: y^2+x^4-x^2 (eight curve), y^4+x^4-1 (square), y^2-x^3-7 (the Bitcoin curve), (x^2+y^2-1)^3+27*x^2*y^2 (astroid).
  
  Or you can draw a random one (of degree 4 over rationals QQ) as follows:
  
  P.<x,y> = PolynomialRing(QQ)
p = P.random_element(degree=4, terms=15)
print p
R.<x,y> = AffineSpace(QQ, 2)
C = Curve(p, R)
C.plot()

  You maybe lucky and get a curve like this one that I got
  
  x^4 - x^2*y^2 + y^4 - x^3 - 5*x^2 + 17/539*y^2 + x - 7/3
  
  For some unexplainable reason I get happy when I get a compact curve.
  
  
• Check if a polynomial of two variables (over complex numbers CC) is reduced (square-free).
  
  P.<x,y>=PolynomialRing(CC)
f = x^4+x*y^3+x^3*y+y^4
gcd([f,f.derivative(x),f.derivative(y)])

  If gcd is not a constant, then the polynomial is not reduced. To find the reduced polynomial replace the last line with
  
  f//gcd([f,f.derivative(x),f.derivative(y)])

  (Working over CC, Sage does not always simplify the constants appearing in the computation. This is not important as we only care about polynomials up to constant factors.)
  
  
• Check if an algebraic curve defined by a polynomial is singular (over complex numbers).
  
  R.<x,y>=AffineSpace(CC, 2)
C = Curve([(x^2+y^2-1)^3+27*x^2*y^2], R)
C.is_singular()
  
  and find the singular points
  
  C.singular_points()
  
  
• This computes the Taylor expansion at a point (x_0,y_0) = (3,-1) up to degree d =10
  
  x,y = var("x y")
taylor(x^2*y + x*y^2 - 4*x*y + x + 9*y, (x,3), (y,-1),10)

  
  
• This empty spot is waiting for you to submit code that computes the multiplicity and the tangent cone of a singularity.



• This empty spot is waiting for you to submit code that expresses a homogeneous polynomial in two variables as a product of linear forms.



• Here is how you draw the Newton polytope of a polynomial
  
  R.<x,y>=AffineSpace(CC, 2)
p = CC(3+4*I)*x^2 + CC(1+I)*x*y + y + x - x^2*y + 3*y^3
P= p.newton_polytope() ; P
  
  and here is how you can see the Minkowski sum of two polyhedra
  
  X = Polyhedron(vertices=[(0,1),(1,0),(0,0),(1,1)])
Y = Polyhedron(vertices=[(0,0), (0,1), (2,3)])
X+Y

  
  
• The following code determines whether a bivariate square-free polynomial p is irreducible over the complex numbers. The number of irreducible components of p is given by the output of [2].
  This code uses quite sophisticated techniques that go beyond the scope of this course but it could be useful as a blackbox which is why it is included here.
  
  singular.lib('deRham.lib')
R = singular.ring('0','(x,y)','dp')
p = singular('x^6+x^2*y^4+y^2*x^4+y^6')
L = singular.list([p])
singular.deRhamCohomology(L);

  
  
• Compute the projectivization of an affine curve
  
  R.<x,y>=AffineSpace(CC, 2)
C = Curve(y - x^3 + x - 1, R)
C.projective_closure()

  
  
• Compute the affine patch of a projective curve
  
  P.<x,y,z> = ProjectiveSpace(CC, 2)
C = Curve(x^3 - x^2*y + y^3 - x^2*z, P)
C.affine_patch(0)

  Use C.affine_patch(1) or C.affine_patch(2) for other patches.
  
  

sagecell