INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

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44 Mezzettithere exists F such that F(P) ≠ 0 and V (F) ⊃ Y ′ . So P ∈ Y \ V (F) ⊂ Y \ Y ′and Y \ V (F) is an affine open neighbourhood of P in Y \ Y ′ = X 0 ⊂ X.□6. The Veronese maps.Let n, d be positive integers; put N(n, d) = ( ) (n+dd − 1. Note that n+d)d isequal to the number of (monic) monomials of degree d in the variables x 0 , . . ., x n ,that is equal to the number of n+1–tuples (i 0 , . . ., i n ) such that i 0 +...+ i n = d,i j ≥ 0. Then in P N(n,d) we can use coordinates {v i0 ...i n}, where i 0 , . . ., i n ≥ 0 andi 0 + . . . + i n = d. For example: if n = 2, d = 2, then N(2, 2) = ( )42 − 1 = 5. In P5we can use coordinates v 200 , v 110 , v 101 , v 020 , v 011 , v 002 .For all n, d we define the map v n,d : P n → P N(n,d) such that [x 0 , . . ., x n ] →[v d00...0 , v d−1,10...0 , . . ., v 0...00d ] where v i0 ...i n= x i 00 x i 11 . . .x i nn : v n,d is clearly a morphism,its image is denoted V n,d and called the Veronese variety of type (n, d). Itis in fact the projective variety of equations:(∗){v i0 ...i nv j0 ...j n− v h0 ...h nv k0 ...k n, ∀i 0 + j 0 = h 0 + k 0 , i 1 + j 1 = h 1 + k 1 , . . .We prove this statement in the particular case n = d = 2; the general case issimilar.First of all, it is clear that the points of v n,d (P n ) satisfy the system (∗).Conversely, assume that P[v 200 , v 110 , . . .] ∈ P 5 satisfies the equations (∗), whichbecome:⎧v 200 v 020 = v1102v ⎪⎨ 200 v 002 = v1012v 002 v 020 = v0112v 200 v 011 = v 110 v 101⎪⎩ v 020 v 101 = v 110 v 011v 110 v 002 = v 011 v 101Then, at least one of the coordinates v 200 , v 020 , v 002 is different from 0.Therefore, if v 200 ≠ 0, then P = v 2,2 ([v 200 , v 110 , v 101 ]); if v 020 ≠ 0, thenP = v 2,2 ([v 110 , v 020 , v 011 ]); if v 002 ≠ 0, then P = v 2,2 ([v 101 , v 011 , v 002 ]). Note that,if two of these three coordinates are different from 0, then the points of P 2 foundin this way have proportional coordinates, so they coincide.We have also proved in this way that v 2,2 is an isomorphism between P 2 andV 2,2 , called the Veronese surface of P 5 . The same happens in the general case.If n = 1, v 1,d : P 1 → P d takes [x 0 , x 1 ] to [x d 0 , xd−1 0 x 1 , . . ., x d 1]: the image iscalled the rational normal curve of degree d, it is isomorphic to P 1 . If d = 3, wefind the skew cubic.Let now X ⊂ P n be a hypersurface of degree d: X = V P (F), withF =∑i 0 +...+i n =da i0 ...i nx i 00 . . .x i nn .
Then v n,d (X) ≃ X: it is the set of points{v i0 ...i n∈ P N(n,d) |Introduction to algebraic geometry 45∑i 0 +...+i n =da i0 ...i nv i0 ...i n= 0 and [v i0 ...i n] ∈ V n,d }.It coincides with V n,d ∩ H, where H is a hyperplane of P N(n,d) : a hyperplanesection of the Veronese variety.The Veronese surface V of P 5 enjoys a lot of interesting properties. Most ofthem follow from its property of being covered by a 2-dimensional family of conics,which are precisely the images via v 2,2 of the lines of the plane.To see this, we’ll use as coordinates in P 5 w 00 , w 01 , w 02 , w 11 , w 12 , w 22 , so thatv 2,2 sends [x 0 , x 1 , x 2 ] to the point of coordinates w ij = x i x j . With this choice ofcoordinates, the equations of V are obtained by annihilating the 2 × 2 minors ofthe symmetric matrix:⎛M = ⎝ w ⎞00 w 01 w 02w 01 w 11 w 12⎠w 02 w 12 w 22Let l be a line of P 2 of equation b 0 x 0 + b 1 x 1 + b 2 x 2 = 0. Its image is the set ofpoints of P 5 with coordinates w ij = x i x j , such that there exists a non-zero triple[x 0 , x 1 , x 2 ] with b 0 x 0 +b 1 x 1 +b 2 x 2 = 0. But this last equation is equivalent to thesystem:⎧⎨b 0 x 2 0 + b 1x 0 x 1 + b 2 x 0 x 2 = 0b⎩ 0 x 0 x 1 + b 1 x 2 1 + b 2x 1 x 2 = 0b 0 x 0 x 2 + b 1 x 1 x 2 + b 2 x 2 2 = 0It represents the intersection of V with the plane(∗){ b0 w 00 + b 1 w 01 + b 2 w 02 = 0b 0 w 01 + b 1 w 11 + b 2 w 12 = 0 ,b 0 w 02 + b 1 w 12 + b 2 w 22 = 0so v 2,2 (l) is a plane curve. Its degree is the number of points in its intersectionwith a general hyperplane in P 5 : this corresponds to the intersection in P 2 of lwith a conic (a hypersurface of degree 2). Therefore v 2,2 (l) is a conic.So the isomorphism v 2,2 transforms the geometry of the lines in the planein the geometry of the conics on the Veronese surface. In particular, given twodistinct points on V , there is exactly one conic contained in V and passing throughthem.From this observation it is easy to deduce that the secant lines of V , i.e. thelines meeting V at two points, are precisely the lines of the planes generated bythe conics contained in V , so that the (closure of the) union of these secant lines
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