Some search on the internet and this site didn't result in any topic about this question of Silverman's The Arithmetic of Elliptic Curves:Let W ⊂ P n be a smooth algebraic set, each of whose component varieties has dimension n − 1. Prove that W is a variety.Hint: use Krull's Hauptidealsatz to show W is the zeroset of a single homogenous polynomial.my ideas: So if I(V) contains at least two linear independent polynomials, we have to prove the height of I(V) is at least 2.After that, it is enough to find one singular point, where we assume I ( V ) = f g with f,g nonconstant polynomials.

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Some search on the internet and this site didn&#039;t result in any topic about this question of Silverman&#039;s The Arithmetic of Elliptic Curves:Let   W  ⊂            P      n       be a smooth algebraic set, each of whose component varieties has dimension   n  −  1. Prove that W is a variety.Hint: use Krull&#039;s Hauptidealsatz to show W is the zeroset of a single homogenous polynomial.my ideas: So if I(V) contains at least two linear independent polynomials, we have to prove the height of I(V) is at least 2.After that, it is enough to find one singular point, where we assume   I  (  V  )  =  f  g with f,g nonconstant polynomials.

dilettato5t1 · Accepted Answer

Step 1If W contains more than one component, then it is the union of projective subvarieties of codimension 1, which necessarily have non-empty intersection. (This is immediate from Bezout but if you prefer a smaller hammer, also follows from the Krull height theorem.) Then, we need to check that W cannot possibly be smooth at a point lying on two or more of these components. We can work affine-locally and assume the point   x  =  (  0  ,  0  ,  …  ,  0  )  ∈            A        n   lies on multiple components, all of codimension 1; by Hauptidealsatz these components are (possibly after passing to a smaller open subset) cut out by single equations. Hence we need to check if that f,g vanish at x, then the hypersurface V(fg) is not smooth at x; indeed all of the partials take the form       f    ′    (  x  )  g  (  x  )  +  f  (  x  )      g    ′    (  x  )  =  0 (slightly abusing notation here).Step 2So the conclusion is that W can only have one component, hence it is an irreducible variety.

Some search on the internet and this site didn't result in any topic about this question of Silverman's The Arithmetic of Elliptic Curves: Let W subset Pn be a smooth algebraic set, each of whose component varieties has dimension n-1. Prove that W is a variety.

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