# Join of two intersecting varieties

Suppose I have two smooth projective varieties $$X$$ and $$Y$$ in $$\mathbb{P}^n$$, that intersect along a smooth subvariety $$Z$$. Is there a formula to compute the degree of the join variety $$J(X,Y)$$ of $$X$$ and $$Y$$? Of course if $$Z$$ is $$\emptyset$$ then we have the classical formula, but what if $$Z$$ is non-empty?

• The join $J(X,Y)$ is the (linear) projection of the abstract join of $X$ and $Y$ in $\mathbb{P}^{2n+1}$, so the question reduces to the question --- how is the degree of a variety related to the degree of its linear projection. – Sasha Aug 5 at 10:36

You can find a lot of material on the intersection theory of join varieties (including all that I write in this answer) in the conference paper [2] of Flenner. At the end of the second page (p.130) you find the statement of the general formula for the degree of joins: $$\deg X \deg Y = \deg V + \deg \pi \deg J,$$ The notation is as follows:

• $$X,Y\subseteq \mathbb P^n$$ are projective (irreducible) subvarieties, over an arbitrary field $$K$$;
• $$V$$ is their Vogel-cycle, which is a special cycle supported on $$X\cap Y$$;
• $$J\subseteq \mathbb P^n$$ is their embedded join;
• $$\pi:\tilde J \to J$$ is the (rational) projection from the abstract join $$\tilde J\subseteq \mathbb P^{2n+1}$$ of $$X$$ and $$Y$$.

This formula is a consequence of the correspondence principle of van Gastel [3] and of the intersection theory of Stückrad and Vogel [4]. Remarkably, it is valid regardless of the dimension of $$J$$. The paper of Flenner also explains how to express the Vogel-cycle (and so also the degree of $$J$$) in terms of Segre classes, cf [1].

1. Ådlandsvik B.: Joins and higher secant varieties. Mathematica Scandinavica (1987) 61:213-222.
2. Flenner H.: Join Varieties and Intersection Theory pp.129-197. In: Ellingsrud G., Fulton W., Vistoli A. (eds) Recent Progress in Intersection Theory. Trends in Mathematics, Birkhäuser 2000.
3. Gastel, L.J. van: Excess intersections and a correspondence principle. Inventiones Mathematicae (1991) 103:197–221.
4. Stückrad J., Vogel, W.: An algebraic approach to the intersection theory pp.1-32. In: The curves seminar at Queen's Vol II. Queen's papers in pure and applied mathematics No.61, Kingston 1982.