Math 130 Midterm 1 Spring, 2012 Professor Hopkins

Transcription

1 Math 130 Midterm 1 Spring, 2012 Professor Hopkins Name: Total Make sure your name is written on every page of the exam.

2 Here are the axioms and some consequences of them for projective planes and projective spaces. You may use these in your proofs, referring to them by number. Definition 1. A projective plane is a set 1 called the set of points and a collection of subsets of 1 called lines satisfying P1-P4 below. P1. Given two distinct points, there is a unique line containing them both. P2. Any two lines meet in at least one point. P3. There exist three non-collinear points. P4. Every line contains at least three points. Definition 2. A projective 3-space is a set whose elements are called points, together with certain subsets called lines, and certain other subsets called planes, which satisfies the following axioms: S1. Two distinct points lie on a unique line. S2. Three non-collinear points lie on a unique plane. S3. A line meets a plane in at least one point. S4. Two planes have at least a line in common. S5. There exist four non-coplanar points, no three of which are collinear. S6. Every line has at least three points. The following are consequences of these axioms. S7. Two distinct planes meet in exactly one line. S8. If A and B are any two distinct points of a projective 3-space then the line AB is the intersection of two distinct planes. S9. If two distinct points lie on a plane, so does the line containing them. S10. A plane and a line not contained in the plane meet in exactly one point. S11. Two distinct concurrent lines are contained in exactly one plane. S12. A line and a point not on it lie on a unique plane. S13. A plane in a projective 3-space contains 3 non-collinear points. S14. A plane in a projective 3-space is a projective plane.

3 1. (10 pts) Prove Desargues Theorem in the special case in which the vertices of one of the triangles are actually collinear.

4 2. (5 pts each) In the figures below, the line l is the line at infinity. Redraw each figure in Euclidean plane, so that the line l is actually at infinity.

5 3. (5 pts each) Let F be a field. As in the lecture notes, it will be convenient to write (s,t) for the point [s,t,1] P 2 (F). (a) Express the condition that [x,y,z] P 2 (F) lies on a line containing [a,b,c] and [a,b,c ] in terms of determinants. (b) Do the same thing for a plane in P 3 (F) containing 3 points.

6 (c) In P 2 (F), what is the ideal point of the line l a,b,c? (d) What are the homogeneous coordinates of the ideal point of the line through (s,t) and (s,t )? (e) What are the homogeneous coordinates of the point of intersection of the line at infinity in P 2 (F) and the line containing the origin and a point (s,t)?

7 4. (10 pts) Let A, A, B, B be four distinct points, and write l = AB, l = A B. Suppose furthermore that none of the four points is equal to l l. Show that there is a unique perspectivity l(a,b) P l (A,B ).

8 5. (5+10 pts) This problem concerns harmonic points. (a) Are the real numbers 0, 10, 12, 15, regarded as points of the x-axis in RP 2 harmonic? (Don t you wish you d let me do this in class? Was I right, or was Coxeter right?) (b) Let F be a field whose characteristic is not 2. Show that 4 collinear points in P 2 (F) are harmonic if and only there is a coordinate system in which they are [1,0,0], [0,1,0], [1,1,0] and [1, 1,0].

9 6. (10+5 pts) Show that in the Fano projective plane, given 3 non-collinear points (P,Q,R) there is a unique S so that P, Q, R, and S are in general position. How many possible frames of reference are there for the Fano projective plane?

10 7. ( pts) The figure below depicts the 13-point projective plane P 2 (F 3 ), drawn as an extended affine plane. As in the homework problem, the lines in the 3 3 grid are the ones which look like lines (though they wrap around), and the 4 remaining points are the ideal points of the lines to which they are connected by a dashed line. Four of the points have been labeled in a new coordinate system. (a) Locate and label the point [1,1,0], [1,0,1] and [0,1,1]. (b) Having done the first part, every line should now have 3 of its points labeled. The fourth is uniquely determined. Locate and label the point [2, 0, 1].

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