Exercises in bold are suggestions for topics for the obligatory assignment presentations on Thursdays. The other exercises are for training purposes. If you have another topic suggestion for the obligatory assignment presentation, please get in touch!
Chapter 1
- Affine normal rational curves. Exercise 1.11 (suggestion: show a picture for d=2 and d=3) DONE
- Cones and homogeneous polynomials. Exercise 1.8 and 1.9. DONE
Some training exercises: 1.1, 1.2, 1.5, 1.6, 1.7
Chapter 2
- Show that the space of nxn matrices with zero determinant is an irreducible algebraic variety (I.e. present Example 2.4 and a solution to Exercise 2.1). Discuss the set W_r of matrices of rank r. DONE
- Show that the space of quadratic forms in n variables of rank at most r is irreducible (Proposition 2.48) and solve Exercise 2.17
Some training exercises: 2.4, 2.6, 2.9, 2.10, 2.11
Chapter 3
Some training exercises: 3.4, 3.5, 3.7, 3.19, 3.20, 3.21, 3.22
Chapter 4
Projective rational normal curve - exercises 4.15 and 4.17.
Affine and projective rational normal curves - homogenise the generators of the ideal of the affine twisted cubic and find its variety, then find a homogeneous ideal defining the projective rational normal curve (homogenise the ideal) and prove that this is the correct ideal.
Some training exercises: 4.1, 4.2, 4.3, 4.4, 4.13, 4.14
Chapter 5
Show that the ideal of the image of the Segre embedding is generated by 2x2 minors of a matrix and that this ideal is prime (see Propositions 5.19 and 5.24)
Formulate and prove a Nullstellensatz for an arbitrary product of projective spaces (Exercise 5.9)
Show that the space of polynomials of degree d which are a d-th power of a linear form is projectively equivalent to a Veronese map, with some requirements on the characteristic of the field (Exercise 5.20)
Some training exercises: 5.10, 5.13, 5.15, 5.19, 5.20
Chapter 7
Prove Propositions 7.53 and 7.56 on the (co)dimensions of intersections in affine and projective space
Some training exercises: 7.1, 7.2, 7.3, 7.7, 7.11, 7.23