next | previous | forward | backward | up | index | toc

# coneOfLines -- cone of lines on a subvariety passing through a point

## Synopsis

• Usage:
coneOfLines(X,p)
coneOfLines(p,X)
• Inputs:
• X, , a subvariety of $\mathbb{P}^n$
• p, , a point on $X$
• Outputs:
• , the subscheme of $\mathbb{P}^n$ consisting of the union of all lines contained in $X$ and passing through $p$

## Description

In the example below we compute the cone of lines passing through the generic point of a smooth del Pezzo fourfold in $\mathbb{P}^7$.

 i1 : K := frac(QQ[a,b,c,d,e]); t = gens ring PP_K^4; phi = rationalMap {minors(2,matrix{{t_0,t_1,t_2},{t_1,t_2,t_3}}) + t_4}; o3 : MultirationalMap (rational map from PP^4 to PP^7) i4 : X = image phi; o4 : ProjectiveVariety, 4-dimensional subvariety of PP^7 i5 : ideal X 2 2 o5 = ideal (y - y y + y y , y y - y y + y y , y - y y + y y , y y - 5 4 6 2 7 4 5 3 6 1 7 4 3 5 0 7 2 4 ------------------------------------------------------------------------ y y + y y , y y - y y + y y ) 1 5 0 6 2 3 1 4 0 5 o5 : Ideal of frac(QQ[a..e])[y ..y ] 0 7 i6 : p := projectiveVariety minors(2,(vars K)||(vars ring PP_K^4)) o6 = point of coordinates [a/e, b/e, c/e, d/e, 1] o6 : ProjectiveVariety, a point in PP^4 i7 : coneOfLines(X,phi p) o7 = surface in PP^7 cut out by 6 hypersurfaces of degrees 1^3 2^3 o7 : ProjectiveVariety, surface in PP^7 i8 : ideal oo 2 -d 2c -b - c + b*d -d c o8 = ideal (y + --*y + --*y + --*y + ----------*y , y + --*y + -*y + 2 e 4 e 5 e 6 2 7 1 e 3 e 4 e ------------------------------------------------------------------------ b -a - b*c + a*d -c 2b -a -*y + --*y + -----------*y , y + --*y + --*y + --*y + e 5 e 6 2 7 0 e 3 e 4 e 5 e ------------------------------------------------------------------------ 2 2 - b + a*c 2 d -2c b c - b*d 2 ----------*y , y - y y + -*y y + ---*y y + -*y y + --------*y , 2 7 5 4 6 e 4 7 e 5 7 e 6 7 2 7 e e ------------------------------------------------------------------------ d -c -b a b*c - a*d 2 2 y y - y y + -*y y + --*y y + --*y y + -*y y + ---------*y , y - 4 5 3 6 e 3 7 e 4 7 e 5 7 e 6 7 2 7 4 e ------------------------------------------------------------------------ 2 c -2b a b - a*c 2 y y + -*y y + ---*y y + -*y y + --------*y ) 3 5 e 3 7 e 4 7 e 5 7 2 7 e o8 : Ideal of frac(QQ[a..e])[y ..y ] 0 7