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I.2.7 depends on I.2.6 so has H = 2 and V = 4; (I.2.6>I.1.10>I.1.6). I.2.8 can be solved with I.2.7 which would give it H = 2 and V = 5. I.2.9 uses I.1.2 so has H = V =2. I.2.10 also depends on I.2.6 so H = 2, V = 4.
- part a) is a direct consequence of I.2.6
- for part b) recall the notation is the intersection of with the standard affine when
- Argue which in turn is equal to
- this can be proved without I.2.7 arguing similarly to how the analogous affine statement is proved, but with I.2.7 have:
where is irreducible. Then homogenizing gives the result.
- is the dehomoginization operator and is the homogenization operator. Let is considered as just an affine variety. Conclude the result from
- for the next part, use the result of I.1.2: . Then check (groebner basis, maybe) that
- part a is straightforward, and part b is a consequence of part a
- part c is a consequence of I.2.6 because where is the affine coordinate ring.