Special linear systems and syzygies
Tóm tắt
LetX be the base locus of a linear system L of hypersurfaces in ℙ
r
(ℂ). In this paper it is showed that the existence of linear syzygies for the ideal ofX has strong consequences on the fibres of the rational map associated toL. The case of hyperquadrics is particularly addressed. The results are applied to the study of rational maps and to the Perazzo’s map for cubic hypersurfaces.
Tài liệu tham khảo
A. Alzati and F. Russo, Some extremal contractions between smooth varieties arising from projective geometry,Proc. London Math. Soc. (3)89 (2004), 25–53.
D.A. Cox, Equations of parametric curves and surfaces via syzygies,Contemp. Math. 286 (2000), 1–20.
L. Degoli, Sui sistemi lineari di quadriche riducibili ed irriducibili a Jacobiana identicamente nulla,Collect. Math. 35 (1984), 131–148.
W. Decker, L. Ein, and F.O. Schreyer, Construction of surfaces in ℙ4,J. Algebraic Geom. 2 (1993), 185–237.
G. Fano, Sulle forme cubiche dello spazio a cinque dimensioni contenenti rigate razionali del 4° ordine,Comment. Math. Helv. 15 (1943), 71–80z.
K. Hulek, S. Katz, and F.O. Schreyer, Cremona transformations and syzygies,Math. Z. 209 (1992), 419–443.
P. Ionescu, Embedded projective varieties with small invariants III,Algebraic Geometry (L’Aquila 1988) 138–154, Lecture Notes in Math.1417, Springer, Berlin, 1990.
J.M. Landsberg, On degenerate secant and tangential varieties and local differential geometry,Duke Math. J. 85 (1996), 605–634.
U. Perazzo, Sulle varietà cubiche la cui hessiana svanisce identicamente,Atti R. Acc. Lincei (1900), 337–354.
R. Permutti, Su certe classi di forme a hessiana indeterminata,Ricerche Mat. 13 (1964), 97–105.
T.G. Room,The Geometry of Determinatal Loci, Cambridge University Press, London, 1938.
F. Russo,Tangents and Secants of Algebraic Varieties, Publicações Matemáticas do IMPA24, Rio de Janeiro, 2003.
F. Russo and A. Simis, On birational maps and Jacobian matrices,Compositio Math. 126 (2001), 335–358.
A. Simis, Cremona transformations and some related algebras,J. Algebra 280 (2004), 162–169.
J.G. Semple and J.A. Tyrrel, TheT 2,4 ofS 6 defined by a rational surface3 F 8,Proc. London Math. Soc. (3)20 (1970), 205–221.
P. Vermeire, Some results on secant varieties leading to a geometric flip construction,Compositio Math. 125 (2001), 263–285.
F.L. Zak,Determinants of Projective Varieties and Their Degrees, Encyclopaedia Math. Sci.132 Springer, Berlin, 2004.