Vrije Universiteit Brussel Academic year 2010-2011
Faculteit Wetenschappen 1st session
3de Bachelor Wiskunde June 24, 2011
Algebraic Geometry Exam โ Exercises
1. Let Rbe a domain with quotient field K, and let Lbe a finite algebraic extension of K.
(a) For any vโL, show that there is a non-zero aโRsuch that av is integral over R.
(b) Show that there is a basis v1, v2, ..., vnfor Lover K(as a vector space) such that each viis
integral over R.
2. Find an irreducible polynomial in R[X, Y ]such that V(F)is reducible.
3. Decompose V(XY 2+Y2X2โX3โY4, X 3+Y5โX2Y2โXY 3)โA2(C)into irreducible
components.
4. Find all points of intersection and the intersection numbers at these points for the following
projective curves:
F= (X3+Y3)Z+X4+Y4
G=X4+Y4โ3X2Y Z โ3X Y 2Z
We work over C.
5. (a) If ฯ:VโWis an onto polynomial map, and Xis an algebraic subset of W, show that
ฯโ1(X)is an algebraic subset of V. If ฯโ1(X)is irreducible show that Xis irreducible.
(b) Show that V(XZ โY2, Y Z โX3, Z 2โX2Y)โA3(C)is a variety.
Exam duration: 3 hours. The use of any course notes is allowed. Good luck!