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IfX is irreducible, prove thatu(X)is irreducible Problem 2

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Duke University – Spring 2017 – MATH 627 January 24, 2017

Problem set 1 Olivier Debarre Due Thursday February 2, 2017

Problem 1. Recall that a topological spaceX isirreducibleif it is non-empty and is not the union of two strict closed subsets. In other words, ifX1 andX2are closed subsets ofXandX =X1∪X2, thenX =X1orX =X2.

a) LetX be a topological space and let V ⊂ X be a subset (endowed with the induced topology).

Prove thatV is irreducible if and only if its closureV is irreducible.

b) LetX andY be topological spaces and letu :X →Y be a continuous map. IfX is irreducible, prove thatu(X)is irreducible

Problem 2. Let k be an infinite (not necessarily algebraically closed) field. Let C ⊂ k2 be the vanishing setV(X2−Y3).

a) Prove that the ideal ofCis the ideal ink[X, Y]generated byX2−Y3 and thatC is irreducible (Hint: use the “parametrization”k→ C given byt 7→ (t3, t2)and express A(C) = k[X, Y]/I(C) as a subring ofk[T]).

b) Prove thatCis not isomorphic tok(Hint: prove thatA(C)is not a principal ideal domain).

c) How do these these results generalize to the vanishing setV(Xr−Ys), whererandsare relatively prime positive integers?

Problem 3. Letkbe aninfinite(not necessarily algebraically closed) field, letu: P1k →P3kbe the regular map defined byu(s, t) = (s3, s2t, st2, t3), and setC :=u(P1k).

a) Prove that no 4 distinct points ofC are contained in a hyperplane inP3k.

b) Prove that any quadric inP3k(i.e., any subset ofP3kdefined by a non-zero homogoneous polyno- mial of degree 2) that contains 7 distinct points ofC containsC.

c) Prove thatCis the vanishing set inP3kof the (homogeneous) idealI ink[T0, T1, T2, T3]generated by the homogeneous polynomialsT0T2−T12, T22−T1T3, T1T2−T0T3, which can be neatly expressed as the2×2-minors of the matrix

T0 T1 T2 T1 T2 T3

.

d) Prove that the ideal ofCisI (Hint: prove that any polynomialP ∈k[T0, T1, T2, T3]is congruent moduloI to a polynomial of the typeA(T0, T1, T3) +T2B(T3)and that ifP vanishes onC, one has B = 0; then, use a similar method to show thatAis divisible byT13−T02T3).

e)(Extra credit)How do these results generalize to the regular mapu: P1k →Pnk(n ≥ 3) defined byu(s, t) = (sn, sn−1t, . . . , stn−1, tn)?

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