# On the equations of the k-secant of the Segre variety

The Segre map

On first hand, the Segre map is the function

$$\sigma_{n,m} : \mathbb{P}^n \times \mathbb{P}^m \rightarrow \mathbb{P}^{(n+1)(m+1)-1} : ([x_0,...,x_n],[y_0,...,y_m]) \rightarrow [x_0y_0, x_0y_1,...,x_0y_m,x_1y_0,...,x_iy_j,...,x_ny_m]$$

and the Segre variety $$\Sigma_{n,m}$$ is the image of the Segre map $$\Sigma_{n,m}=\sigma_{n,m}(\mathbb{P}^n \times \mathbb{P}^m).$$

Let $$Z_{i,j}=x_iy_j$$. The Segre variety is also defined as the common zero locus of the quadratic polynomials $$Z_{{i,j}}Z_{{k,l}}-Z_{{i,l}}Z_{{k,j}},$$ where $$Z_{i,j}$$ is understood to be the natural coordinate on the image of the Segre map.

The k-secant of a projective variety

The k-secant of $$X\subset \mathbb{P}^n$$ is $$\sigma_k(X)=\overline{\underset{x1,...,x_k\in X}{\cup} \mathbb{P}^{k-1}_{x_1,...,x_k}}$$ where $$\mathbb{P}^{k-1}_{x_1,...,x_s}$$ is a projective space of dimension $$s-1$$ passing through $$x_1,...,x_s$$. If needed to better understand this definition, another equivalent approach to define k-secant is the following.

Let $$Y\subset \mathbb{P}^n$$. The joint of X and Y is $$J(X,Y)=\overline{\underset{x\in X, y\in Y, x\neq y}{\cup} \mathbb{P}^1_{x,y}}$$ where $$\mathbb{P}^1_{x,y}$$ is the projective line containing x and y.

The 2-secant of X, also simply named the secant of X, is $$\sigma(X)=J(X,X)$$ and the k-secant is $$\sigma_k(X)=J(X,\sigma_{k-1}(X)).$$

Question

What is the set of polynomials whose common zero locus is the k-secant of the Segre variety ?

Consider the matrix of size $$(m+1)$$ by $$(n+1)$$ with entries $$x_iy_j$$. Then the equations of the $$k$$-secant variety are all minors of this matrix of size $$k + 1$$.
• The main idea is to think of the ambient space as of the space of matrices; then the $k$-secant variety is the variety of matrices of rank $k$. Commented Feb 17, 2022 at 19:54