By H. A. Nielsen

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**Additional resources for Algebraic varieties [Lecture notes]**

**Example text**

Let X = V (I) ⊂ An be an affine set. For x ∈ X and f ∈ I the differential n df = 1 ∂f (x)(Xi − xi ) ∂Xi is a linear polynomial in k[X1 , . . , Xn ]. The (embedded) tangent space to X at x is the affine linear subspace Tx X = V ({df |f ∈ I}) Let X = V (I) ⊂ Pn be a projective set. For x ∈ X and F ∈ I homogeneous the differential n ∂f dF = (x)Xi ∂X i 0 is a homogeneous polynomial in k[X0 , . . , Xn ]. The projective (embedded) tangent space to X at x is the projective linear subspace Tx X = V ({dF |F ∈ I}) n ∂f 0 ∂Xi Xi = deg(F ) F it follows that the restriction of the projective By Euler’s formula tangent space to an open affine coordinate space in Pn is identified with the affine tangent space.

Then f : X → f (X) is an isomorphism. Proof. 1 f ∗ : OY,y → OX,x is surjective for all x. Assume X, Y affine and let J be the kernel of f ∗ : k[Y ] → k[X]. This gives k[Y ]/J k[X]. 10. Let X be irreducible of dim X = n. Then there is a nonempty open subset U ⊆ X which is isomorphic to an open subset of an irreducible hypersurface in An+1 . Proof. 2 regular functions f1 , . . , fn such that k[X] is integral over the polynomial ring k[f1 , . . , fn ] and k(f1 , . . , fn ) ⊂ k(X) is separable. 1.

Let 0 ≤ j0 , . . , jm ≤ n + 1 be indices. jm = 0. jσ(m) . jm r for all indices and 0 ≤ s ≤ m. 4. Lines in P3 , n = 3, m = 1 give one nontrivial relation y01 y23 − y02 y13 + y03 y12 = 0 3,1 defining G as a hypersurface in P5 . 5. By Gauss elimination the Grassmann variety is covered by open affine spaces A(m+1)(n−m) . It follows that Gn,m is an irreducible nonsingular projective variety of dimension dim Gn,m = (m + 1)(n − m) 19. 1. Let X ⊂ PN be a nonsingular projective variety of dimension n. Let W ⊂ PN × PN × PN be points on lines W = {(x, y, z)|z on the line through x, y} Let ∆X ⊂ X × X be the diagonal.