CHAPTER 7

Projective Geometry over a Finite Field

Joseph A. THAS

Department of Pure Mathematics and Computer Algebra, University of Ghent, Krijgslaan 281, B-9000 Gent, Belgium

Contents

Introduction

1. k-arcs and normal rational curves

1.1. Definitions

1.2. The three problems of B. Segre

1.3. k-arcs in

1.4. k-arcs in

1.5. k-arcs in

1.6. The nonclassical 10-arc of

1.7. The nucleus or kernel of a normal rational curve, and -arcs in

1.8. The duality principle for k-arcs

1.9. Open problems

2. k-arcs and MDS codes

2.1. MDS codes

2.2. The general case

2.3. Linear MDS codes

3. k-caps and ovoids

3.1. Definitions

3.2. k-caps in

3.3. Ovoids and inversive planes

3.4. Caps in ovoids

3.5. Maximum size of a cap in

3.6. Open problems

4. Maximal arcs and Hermitian arcs

4.1. Maximal arcs

4.2. Maximal arcs in with q odd

4.3. Maximal arcs in with q even

4.4. Hermitian arcs and Hermitian curves

4.5. Characterizations of nonsingular Hermitian curves

4.6. Hermitian arcs other than nonsingular Hermitian curves

4.7. A characterization theorem of Tallini Scafati

4.8. Open problems

5. Semi-ovals, semi-ovoids and two combinatorial characterizations of ovoids

5.1. A combinatorial characterization of ovoids

5.2. Semi-ovals, semi-ovoids, and a second combinatorial characterization of ovoids

5.3. Regular semi-ovals

6. Sets of type

6.1. Introduction

6.2. The classification for m=1, m=2 and m=q

6.3. Projections of quadrics

6.4. The classification for , , and

6.5. The classification for m=3, and q=4

7. Blocking sets

7.1. Introduction

7.2. Baer cones

7.3. Main results

7.4. Hyperplane coverings and minimal blocking sets

7.5. Open problems

8. Spreads and partial spreads

8.1. t-spreads and partial t-spreads of

8.2. Geometric t-spreads

8.3. t-spreads of

8.4. Spreads and partial spreads of

8.5. Partition of into subgeometries

8.6. Open problems

9. Ovoids and spreads of classical polar spaces, hemisystems

9.1. Finite classical polar spaces

9.2. Ovoids and spreads of polar spaces

9.3. Existence and nonexistence of spreads

9.4. Existence and nonexistence of ovoids

9.5. Hemisystems

10. Flocks, partial flocks and maximal exterior sets

10.1. Flocks

10.2. Flocks, translation planes and generalized quadrangles

10.3. Flocks of ovoids

10.4. Flocks of hyperbolic quadrics

10.5. Flocks of cones

10.6. Maximal exterior sets

10.7. Partial flocks

10.8. Open problems

References