# 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

## 1 Answer

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  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  and of the intersection theory of Stückrad and Vogel . 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. 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.