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Homework on Chapter 1
1.A [UAG ], Ex. 1.1.
1.B Write down the equation of two nonsingular conics Q 1 , Q 2 in P^2 R such that the intersection Q 1 + Q 2 is 2 P 1 + P 2 + P 3 (see [UAG ], (1.12)).
1.C [UAG ], (1.7) says that a nondegenerate conic can b e parametrised.
Discuss the converse: a map P^1! P^2 de ned by
(U; V ) 7! (a 0 (U; V ); a 1 (U; V ); a 2 (U; V ));
where a 0 ; a 1 ; a 2 are linearly indep endent quadratic forms in U; V has image a plane conic. Determine the image if
a 0 = U 2 2 V 2 ; a 1 = U V ; a 2 = U 2 + V 2 :
1.D The following is a section of a past exam pap er. (Please b e very brief on the b o okwork part.) Show that if P 1 ; : : : ; P 4 are 4 p oints of P^2 such that no 3 are collinear
then there is exactly a p encil fQ 1 + Q 2 g of conics in P^2 through them.
Determine all the conics through the 5 p oints
(1; 0 ; 0) (0; 1 ; 0) (0; 1 ; 1) (1; 1 ; 1) (2; 1 ; 3):
Homework on Chapter 2
2.A [UAG ], Ex. 2.1.
2.B Pappus' theorem: let L; M P^2 C b e lines, and supp ose that A; B ; C 2
L and A^0 ; B 0 ; C 0 2 M are distinct p oints. Construct the 6 lines joining
A^0 B ; AB 0 ; A^0 C ; AC 0 ; B 0 C ; B C 0 , and the 3 p oints of intersection
P = B C 0 \ B 0 C ; Q = AC 0 \ A^0 C ; R = AB 0 \ A^0 B :
Then P ; Q; R are collinear. Prove this using Corollary 2.7.
2.C Supp ose that C 1 ; C 2 P^2 C are quartic curves (that is, curves of de-
gree 4) such that
C 1 \ C 2 = fP 1 ; : : : ; P 16 g
(16 distinct p oints). Sketch a plausible argument for the following implica- tion: a quartic curve D passing through P 1 ; : : : ; P 13 also passes through the last 3 p oints P 14 ; P 15 ; P 16.
2.D The following is a section of a past exam pap er. (Please b e very brief on the b o okwork part.)
Supp ose that k C is a sub eld and let C P^2 k b e the nonsingular
cubic curve C : Y 2 = X 3 + aX Z 2 + bZ 3 with O = (0; 1 ; 0). Show how to construct a group law on C with O as origin. Your construction must b e everywhere de ned, and you should explain the construction of the inverse, but need not prove the group axioms. Supp ose C is given in ane co ordinates by C : Y 2 = X 3 + 1 and let
P = (2; 3) 2 C and Q = (0; 1) 2 C. Calculate P + Q in the group law.
Calculate the tangent line to C at P , and using this, show that 2 P = Q. Prove that P generates a subgroup of C of order 6. [Hint: The tangent lines to C at Q is (y = 1).]
Homework on Chapter 4
4.A The central result of [UAG ], Chapter 4 is Prop osition 4.5: Polynomial
maps f : V! W b etween ane algebraic varieties V ; W corresp ond one-to-
one with k -algebra homomorphisms : k [W ]! k [V ]. Let V = A^1 , W = A^3 ,
and f : V! W the map t 7! t; t^2 ; t^3. What is ?
4.B Let k [V ] = k [X ; Y ]=(Y 2 X 2 X ), k [W ] = k [Z ; T ] and : k [W ]!
k [V ] the homomorphism Z 7! X Y ; T 7! X. What is f?
4.C Do [UAG ], Ex. 4.9, and say why it is relevant to the de nition of dom f in 4.7.
4.D Past exam question. Be brief on the b o okwork. (a) De ne the co ordinate ring k [W ] of an irreducible algebraic subset
W An^ and explain how to view it as a ring of functions on W. The
Nullstellensatz for An^ states that the natural corresp ondences I and V de-
termine a bijection b etween irreducible algebraic sets W An^ and prime
ideals of k [X 1 ; : : : ; Xn ]. State and prove the analogous result for irreducible algebraic subsets of W in terms of ideals of k [W ].
(b) Let ': A^1! A^2 b e the map given by
T 7! (T 2 T ; T 3 T ):
Determine the equation f (X ; Y ) = 0 of the image C = '(A^1 ). Prove directly (without using the Nullstellensatz) that any p olynomial in k [X ; Y ] that vanishes on C is divisible by f (X ; Y ). Show that ' has a rational inverse , and determine the set of p oints where is not regular.
Homework on Chapter 5
5.A Consider pro jective space Pnk over an algebraically closed eld k. If
f = f (x 0 ; : : : ; xn ) is a homogeneous p olynomial and P 2 Pn^ a p oint, ex-
plain what the condition f (P ) = 0 means. Let f 1 ; : : : ; fm b e homogeneous p olynomials; de ne the variety
V (f 1 ; : : : ; fm ) Pn
of common zeros of the fi. Prove that V (f 1 ; : : : ; fm ) = ; if and only if
X ia i 2 (f 1 ; : : : ; fm ) for every i and for some ai.
5.B For an irreducible subvariety X Pn^ , de ne the homogeneous co-
ordinate ring k [X ] := k [x 0 ; : : : ; xn ]=I (X ), where I (X ) is the homogeneous
ideal of forms vanishing on X. If Y Pm^ has homogeneous co ordinate ring
k [Y ], nd the necessary and sucient condition that k [X ] = k [Y ] in terms
of the geometry of X Pn^ , Y Pm^. Give an example of two isomorphic
pro jective varieties whose homogeneous co ordinate rings are not isomorphic.
5.C Do [UAG ], Ex. 5.2.
5.D Half of a past exam question.
Let V Pn^ b e an irreducible pro jective variety over an algebraically
closed eld k ; say what is meant by the homogeneous ideal I (V ) of V , and de ne the function eld k (V ) of V. Explain brie y how k (V ) is related to
the co ordinate ring of an ane piece of V. If C P^3 is the curve de ned
by the 3 equations
X Z = Y 2 ; X T = Y Z and Y T = Z 2 ;
prove that k (C ) is isomorphic to the eld k (t) of rational functions in an indeterminate t.
Homework on Chapter 7
7.A Let L S 3 P^3 b e a line on a nonsingular cubic surface. Show that
the linear pro jection from L extends to a morphism S 3! P^1. [Hint: Write
L : (Z = T = 0) and S : (AZ B T = 0) for suitable A; B. At some p oint
you must use the fact that S is nonsingular along L.]
7.B Let S : (X 3 + Y 3 = Z 3 + T 3 ) and L : (X + Y = Z + T = 0). Find the 10 lines meeting L explicitly. [Hint: You can guess the answer by doing [UAG ], Ex. 7.6, or you can work it out using the regular metho d of Prop osition 7.3.]
7.C [UAG ], Ex. 7.1.
7.D Past exam question. Consider the singular cubic surface
S : (X Y Z + X Y T + X Z T + Y Z T = 0):
Find the singular p oints. The 6 lines of P^3 joining two singular p oints lie on S. Find all the lines of S that intersect the line X = Y = 0. Prove that S contains exactly 6 lines.
References
[UAG] M. Reid, Undergraduate algebraic geometry, CUP.
