% Authors: R/"udiger Achilles and Davide Aliffi % achilles@dm.unibo.it, aliffi@dm.unibo.it COMMENT This is an example session demonstrating and testing the facilities offered by the module SEGRE, an extension of the REDUCE package CALI. END COMMENT; comment ################################################## ### ### ### Joins, secant varieties, ### ### relative tangent cones, limits of joins ### ### ### ################################################## end comment; % Example 1. Relative tangent cone. % [T. Krasi{\'n}ski: The join of algebraic curves. Preprint University % of Lodz, 2000], Example 3.8, or T. Krasi{\'n}ski: Relative tangent % cone to analytic curves. Preprint University of Lodz, 2000], % Example 1. % Given A:={(t^2,t^3,0): t \in C} and B:={(t^2,0,t^3): t \in C}, % we calculate the ideal of the relative tangent cone C_0(A,B): setring({x,y,z},{},lex); a:={y**2-x**3,z}; b:={z**2-x**3,y}; rtc:=ljoin(a,b); % If we want only the underlying analytic set of the relative % tangent cone of A and B at O, then we need to compute the radical: setideal(rtc,rtc)\$ radical rtc; % Example 2. Secant variety. % This example has been taken from % [A. Simis and B. Ulrich: On the ideal of an embedded join. % J. Algebra 226 (2000), 1-14], Example 2.4 % or [A. Simis: Algebraic aspects of tangent cones. In: Matem{\'a}tica % Contempor{\'a)nea, Proceedings of the XII Escola de {\'A}lgebra, % D. Avritzer and M. Spira, Eds., Diamantina, Brazil, 1994] % Example 5.8. % Let us consider the projective variety X in P^5 defined by the % following ideal p: setring(rest vars 6,{},lex); setideal(p, {x1*x2-x2*x5+x3*x5-x5*x6,x1*x3+x1*x4+x1*x6+x4*x5,x1*x6+x3*x5, x2*x3+x2*x6-x3*x6+x6**2,x2*x4+x3*x6,x3**2+x3*x4+x3*x6-x4*x6}); isprime p; % Let us compute the secant variety Sec(X) = Join(X,X). ljoin(p,p); % The result means that Sec(X) = P^5. % The initial ideal of p with respect to the lexicographic order % with x1 > x2 > x4 > x5 > x3 > x6 is: setideal(inp,{x1*x2,x1*x4,x1*x6,x2*x3,x2*x4,x3*x4}); % It defines an arithmetically Cohen-Macaulay surface whose secant % variety is non-equidimensional of dimension 4. setideal(sec,ljoin(inp,inp)); primarydecomposition sec; adeg sec; % Example 3. Secant variety of the % Segre embedding X = P^2 x P^2 -> P^8; % see e.g., [H. Flenner, L. O'Carroll, W. Vogel: Joins and % intersections. Springer-Verlag Berlin-Heidelberg, 1999], p. 26-27. % The ideal id of the Segre embedding is generated by the 2 x 2-minors % of a generic 3 x 3-matrix m. setring(vars 8,{},lex); m:=mat((x0,x1,x2),(x3,x4,x5),(x6,x7,x8)); id:=ideal_of_minors(m,2); sec:=ljoin(id,id); % The result shows us that the secant variety is a hypersurface % of degree 3. In fact, its equation is the determinant of m. det m; % Example 4. Secant variety of the Veronesion surface v in P^5; % see op. cit. v:=veronese(2,2); secv:=ljoin(v,v); % The equation of the secant variety is the determinant of a % symmetric matrix s. s:=mat((x0,x1,x2),(x1,x3,x4),(x2,x4,x5)); det s; ;end;