Osculatory behavior and second dual varieties of del Pezzo surfaces

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Lanteri, Antonio ; Mallavibarrena Martínez de Castro, Raquel (2001)
  • Publisher: De Gruyter
  • Subject: Geometria algebraica

Let X be a projective manifold, embedded via a line bundle L. For any point x∈X the k-th projective tangent bundle is defined as Tk,x=P(H0(L/mk+1x)). There is a natural map jk from the global sections H0(L) to Tk,x, assigning to each section its k-th jet jk,x(s) at x. After choosing local parameters x1,…,xn, around the point x, jk,x(s) is given by the t-tuple of coefficients of the Taylor expansion of s around x, truncated at the k-th degree. The k-th osculatory space at k is given by Ok,x=P(im(jk)). When the map jk is onto at x, the line bundle is said to be k-jet ample at x, i.e. the osculatory space Ok,x=Tk,x has maximal rank. It is natural to try to give a characterization of the osculatory spaces to a given variety, polarized by a given line bundle. R. Piene and X. S. Dai [in Enumerative geometry (Sitges, 1987), 215–224, Lecture Notes in Math., 1436, Springer, Berlin, 1990; MR1068967 (91k:14040)] studied the osculatory behavior of balanced rational normal scrolls. D. Perkinson [Michigan Math. J. 48 (2000), 483–515; MR1786502 (2001h:14066)] gave a combinatorial characterization of the rank of the osculatory space for nonsingular toric varieties. In the paper under review the authors analyze the second osculatory behavior of del Pezzo surfaces polarized by the anticanonical line bundle. Del Pezzo surfaces are constructed by blowing up P2 in d points in general positions, for d=0,…,8. For d=0 the second and the third osculatory space are of maximal rank at every point. The authors prove that for d=1,2,3,4 the second osculatory space is generically of maximal rank. The locus where the rank drops to 5 and 4 is given by the union of the exceptional divisors and the proper transforms of lines joining two points blown up. For d=5 it is proven that the rank is never maximal; it is always 5 unless the point belongs to the intersection of an exceptional divisor and a proper transform of a line joining two points blown up. For d=6 the rank is 4 unless the point belongs to the intersection of three coplanar lines of a smooth cubic surface in P3. The second osculatory spaces parametrize hyperplane sections which are singular at x, with multiplicity ≥3. Using the results on the second osculatory spaces, the authors are able to give a detailed description (degree, lower-dimensional components) of the second dual variety
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