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lecturenotes 7,8

# lecturenotes 7,8 - Lectures on algebraic geometry Jacobs...

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Lectures on algebraic geometry Jacobs 2010 Notes of lectures 7 and 8 Alan Huckleberry March 9, 2010 Lecture 7 was primarily devoted to answering questions concerning the linear families of curves that were discussed int Lecture 6. Here we begin by review this material and then going on to the new material on conics which was presented in Lecture 8. We have also added two references to the literature. 1 Linear families Previously we considered two relatively prime homogeneous polynomials P 0 and P 1 of degree d in P 2 which define smooth curves C 0 and C 1 . The linear family F := { ([ ζ 0 : ζ 1 ] , z ) P 1 × P 2 ; ζ 0 P 0 ( z ) + ζ 1 P ( z ) = 0 } . is viewed as connecting C 0 and C 1 . Letting π : F → P 1 be defined by the projection on the first factor, we noted that there exist smooth real curves γ : I P 1 defined on the interval I = [0 , 1] with γ (0) = [1 : 0] and γ (1) = [0 : 1] so that the pullback π : F γ I is a smooth trivial fiber bundle. This shows in particular that C 0 and C 1 are diffeomorphic. In fact they are diffeomorphic to any of the intermediate curves C t . The C t are smooth complex algebraic subvarieties of P 2 . There are are special cases where they do not vary complex analytically or, equivalently, algebraic geometrically. However, as we will see in the next lectures, for d 3 one should expect that these structures vary nontrivially. This type of variation of structure is a major theme in algebraic geometry. 1

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One goal of the above construction was to prove the weaker result that e top ( C 0 ) = e top ( C 1 ). Using an explicit calculation for the Fermat curve C 0 this resulted in the genus formula g ( C ) = ( d - 1)( d - 2) 2 (1) for smooth curves C of degree d . Observe that the family F is defined for any two relatively prime polynomials and that the map F γ P 2 defined by projection on the second factor can be regarded as a 3-cycle in singular homology theory. The boundary of this 3-cycle is C 0 - C 1 . This proves the following observation. Proposition 1.1. Any two curves of degree d in P 2 are homologous. In particular, any curve of degree d is homologous to a smooth curve of genus g = ( d - 1)( d - 2) 2 . Hence we may think of the righhand side of (1) as a homology invariant and therefore it carries its own name, virtual genus π ( C ). Independent of the embedding in P 2 , we can regard this as the genus of any smooth curve homologous to C . 2 Conics The simplest curves in P 2 are of degree d = 1. These are the curves defined by linear functions and as a consequence it is an easy matter to parameterize them. Exercise. Let V be a 3-dimensional complex vector space. Show that the projective space P ( V * ) of the dual space V * parameterizes the curves C of degree d = 1 in P ( V ). A curve of degree two in P 2 is called a conic . If we think of z = [ z 0 : z 1 : z 2 ] as a (homogeneous) column vector, then every homogeneous polynomial P of degree d = 2 can be uniquely written as P ( z ) = z t Sz where S is a 3 × 3 symmetric matrix. Thus we may regard the 5-dimensional projective space P (Sym 3 × 3 ) of the vector space of all symmetric matrices as the space of all conics in P 2 .
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