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Extra info for Lectures on Curves, Surfaces and Projective Varieties (Ems Textbooks in Mathematics)
R s C 2/. f0 C tfj / D n X xi iD0 @ . f C tfj / D zero: @xi zero Arguing by way of contradiction, consider that fj . P / ¤ zero (or, equivalently, f0 . P / ¤ zero) which means P isn't really an . s 1/-fold element of the bottom number of the pencil ˆj . Substituting the price f0 tD fj in (6. three) we discover 'i . x0 ; : : : ; xn / WD in addition to @ @xi f0 fj ! 1 D fj @f0 @xi f0 @fj D zero; fj @xi @f0 @xi f0 @fj fj @xi i D zero; : : : ; n; ! D zero; i D zero; : : : ; n: this suggests that the purpose P doesn't depend upon t (since P satisfies the equation 'i . x0 ; : : : ; xn / D zero, the place the rational functionality 'i is self sufficient of t , i D zero; : : : ; n), and so contradicts the idea made on P . therefore one should have fj . P / D f0 . P / D zero, that is to claim that P is an . s 1/-fold aspect for the bottom number of the pencil ˆj . because the similar reasoning applies to every of the pencils ˆj as j varies in f1; : : : ; hg, this establishes the specified end. allow us to ultimately observe a very important element: 'i . x0 ; x1 ; : : : ; xn / cannot be identically 0 for each i D zero; : : : ; n. in a different way ! f0 @ @xi fj will be identically 0 for each i , and so one could have f0 D kfj for a few ok 2 C. hence, for every i D zero; : : : n, it should stick to that @ @ . f0 / D ok . f /; @xi @xi j and so (6. three) will be corresponding to @[email protected] i . fj / D zero, i D zero; : : : ; n. therefore there wouldn't exist variable s-fold issues for the widespread hypersurface of †. 6. four. Jacobian loci 177 P three with equation instance 6. three. 12. The quadric F x3 / D zero x12 C x0 . x2 is a cone with vertex Œ0; zero; ; 1. As varies it describes a pencil † of quadrics whose popular point has a double aspect. The locus of the double issues Œ0; zero; ; 1 of the quadrics of † is the road x0 D x1 D zero that's a (simple) bottom line for †. 6. four Jacobian loci We give some thought to the linear process † of hypersurfaces Xnr zero f0 C C h fh 1 P n of equation D zero; the place fj are linearly self reliant homogeneous polynomials of measure r, in order that dim † D h. Theorem 6. three. eleven assures us that the everyday hypersurface of † doesn't have singular issues outdoors the bottom kind. There could although be specific hypersurfaces in † having a a number of aspect that's not a base aspect of the approach. allow P D Œy0 ; y1 ; : : : ; yn be some extent of P n that's a number of for a few hypersurface of the approach. Then there exist h C 1 parts zero ; : : : ; h (not all of that are 0) from the sector okay such that zero @f0 C @yi 1 @f1 C @yi C h @fh D zero; @yi i D zero; : : : ; n; (6. four) the place we now have written @[email protected] i instead of @[email protected] i . y0 ; y1 ; : : : ; yn /. The linear homogeneous method (6. four) therefore admits non-trivial options and so the Jacobian matrix @fj (with n C 1 rows and h C 1 columns) has rank < h C 1 at P , that's @xi j D0;:::;h iD0;:::;n Â @fj % @yi Ã < h C 1: (6. five) Conversely, if this holds the approach (6. four) has non-trivial strategies and so there exist hypersurfaces of † having P as no less than a double aspect. The projective style that is the locus of the zeros of the suitable generated by way of the minors of order h C 1 of the Jacobian matrix is named the Jacobian number of †.