# EmbeddedProjectiveVariety ++ EmbeddedProjectiveVariety -- join of projective varieties

## Synopsis

• Operator: ++
• Usage:
X ++ Y
• Inputs:
• Y, , in the same ambient projective space of $X$
• Outputs:
• , the join of $X$ and $Y$, that is, the closure of the union of lines of the form $\langle p,q\rangle$, with $p\in X$, $q\in Y$, and $p\neq q$

## Description

 i1 : K = ZZ/333331; i2 : C = PP_K^(1,5); -- rational normal quintic curve o2 : ProjectiveVariety, curve in PP^5 i3 : L = linearSpan {point ambient C,point ambient C}; -- random line o3 : ProjectiveVariety, curve in PP^5 i4 : C ++ L o4 = threefold in PP^5 cut out by 5 hypersurfaces of degrees 3^4 4^1 o4 : ProjectiveVariety, threefold in PP^5 i5 : C ++ C o5 = threefold in PP^5 cut out by 4 hypersurfaces of degree 3 o5 : ProjectiveVariety, threefold in PP^5 i6 : (point C) ++ (point C) ++ (point C) o6 = surface in PP^5 cut out by 3 hypersurfaces of degree 1 o6 : ProjectiveVariety, surface in PP^5