Algebraic Geometry Sundance 1986 by Holme R. Speiser (Eds.) PDF

By Holme R. Speiser (Eds.)

ISBN-10: 3540192360

ISBN-13: 9783540192367

This quantity offers chosen papers due to the assembly at Sundance on enumerative algebraic geometry. The papers are unique examine articles and focus on the underlying geometry of the topic.

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In the d e f o r m a t i o n space of a tacnode y 2 _ y x 2 + tlx2 + t2 x + t3 = 0 the following loci m a y be described as follows. First, the locus of c u r v e s w i t h two nodes m a y be given p a r a m e t r i c a l l y b y tI = e, t2 = O, t3 = -e 2 or in Cartesian f o r m as t 2 = O, t 2 -- - t 3. The locus of curves with cusps is given parametrically by t l = ~ d 2, t2 = 2d 3, t3 = 3 d 4 or in Cartesian f o r m as t2 = 3t3, 9t2 = 3 2 t i t 3. , the closure of the locus of curves with one node) is given in Cartesian form as -64t 3 - 128t 2t 2 - 27t4 + 144tlt22t 3 - 64t 4t3+16t~t2 and p a r a m e t r i c a l l y as t I = s, t 2 = c3 - 2cs, t 3 = - ~ c 4 + c 2s.

A node such t h a t the t a n g e n t line to one of the b r a n c h e s has c o n t a c t order t h r e e or m o r e with t h a t branch The divisor ]TB of c u r v e s with a flex bitangent - - t h a t is, a bitangent line h a v i n g c o n t a c t of o r d e r t h r e e o r m o r e w : t h the c u r v e at one of its points of tangency the divisor NL of c u r v e s with a node located s o m e w h e r e on a fixed line L c ~2 and m a n y others described in [D-HI]. We can also define divisors on C as well: for example, the divisor N of points of C lying over assigned nodes of the corresponding plane curves, and the divisor F of points lying over flexes.

W(d,S)' 9(d,8) • ~ V'(d,8) ig W(d,8+l) 9(d,8+I) The morphisms nl, n2, n3, and n 4 are normalizations and the morphism g comes from the universal mapping property of normalization applied to n 3 . It is clear t h a t g is proper. W' (d, 8) is obtained from W (d, 8) by adding codimension two subvarieties. 4) below) one m a y see t h a t W' (d, 8) and W (d, 8) are both smooth. This allows us to identify Pic(W(d, 6)) and Pic(W'(d, 8)); call this identification j,. From standard intersection theory (See Fulton, [F]) we get a 30 homomorphism g~n 2 j .

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Algebraic Geometry Sundance 1986 by Holme R. Speiser (Eds.)


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