Mathematics 502
Homework (due Feb 21)
A. Hulpke
15∗ ) Prove Desargues theorem (9.5.3) for PG(2, q). (“Concurrent” means that the lines
a1 a2 b1 b2 and c1 c2 intersect in the same point.)
16) In continuation of problem 13 we consider the near-field (F, +, ?).
a) Show that x ? x = −1 for x 6= 0, 1, 2.
b) Show that (−1) ? x = −x = x + x for x ∈ F.
c) Show that if a, b, c, d ∈ F with a 6= c the equations a ? x + y = b, c ? x + y = d have a
unique solution (x, y).
17) Now let Ω be the projective plane arising from (F, +, ?) via MOLS. One can show
(you may use this without proof which is just tedious checking) that we can take as points
X = {(a, b) | a, b ∈ F} ∪ {(m) | m ∈ F} ∪ {∞}
{z
} |
{z
}
|
affine plane
Points at infinity
with lines
y = x ? m + b containing all points (x, y) fulfilling the equation and (m),
x = a containing all points (a, b) and (∞).
L∞ containing (∞) and all points (m).
We now take the following points:
a1 = (1, 1)
b1 = (−i + 1, i − 1) c1 = (1, i)
a2 = (−i + 1, 1) b2 = (−1, −i + 1)
c2 = (−1, −i − 1)
a) Show that line a1 a2 has equation y = 1, line b1 b2 has equation y = x ? i + 1 and line c1 c2
has equation y = x ? (i − 1) + 1 and that the three lines intersect in (0, 1) — thus we have
two triangles in perspective (i.e. in the book’s language the lines are concurrent).
b) Show that the lines of the triangles fulfill the following equations
a1 b1 : y = x ? (−i + 1) + i a1 c1 : x = 1
b1 c1 : y = x ? (−i) − i
a2 b2 : y = x ? (i − 1)
a2 c2 : y = x ? (−i) + i − 1 b2 c2 : x = −1
c) Deduce that p = b1 c1 ∩ b2 c2 = (−1, 0), q = a1 c1 ∩ a2 c2 = (1, 2), r = a1 b1 ∩ a2 b2 = (i + 1, −i).
d) Show that r does not lie on pq. (Thus Ω is not desarguesian and therefore a different
plane that PG(2, 9)!)
18) a) Prove that if q is a prime power then any five points of PG(2, q), no three collinear,
are contained in a unique conic.
b) Deduce that there are (q2 + q + 1)q2 (q − 1) conics.