T ECHNISCHE U NIVERSIT ÄT B ERLIN
Institut für Mathematik
Geometry I - WS 15
Prof. Dr. John Sullivan
Hana Kouřimská
http://www.math.tu-berlin.de/geometrie/Lehre/WS15/GeometryI/
Exercise Sheet 7
Exercise 1: Projective space.
Let P (V ) be a 3-dimensional projective space. Show that:
(4 pts)
(a) Any two planes in P (V ) intersect in a line.
(b) If A is a plane in P (V ) and l is a line that is not contained in A, then A and l
intersect in exactly one point.
Exercise 2: Separating the projective plane with lines.
(4 pts)
2
Suppose we have three lines in the real projective plane RP , which do not all pass
through a common point. How many regions do they cut the plane into? What if there
are four lines, no three of which pass through a common point? Optional: what about n
lines?
Exercise 3: Projective geometry with different fields.
(4 pts)
3
Let V = Z2 be a vector space of dimension 3 over the two-element field Z2 = Z/2Z
and consider the projective space P (V ). How many points does it contain? How many
lines? How many points lie on each line? How many lines pass through each point?
Exercise 4: A projective triangle.
(4 pts)
3
Let U1 , U2 and U3 be the three 2-dimensional subspaces of R defined by x = 0, x +
y + z = 0 and 3x − 4y + 5z = 0, respectively. Find the vertices of the triangle in P (R3 )
whose sides are the three projective lines P (Ui ).
Due: Before lecture on Tuesday, December 8, 2015.