Defining ideal of the Segre locus in arbitrary characteristic

We give a new proof of the linearity of the Segre locus, that is, the locus of points from which a variety is projected non-birationally. Our proof works in the case where the characteristic is zero or large enough. For small characteristics, we give an example of a variety whose Segre locus is non-linear. To show these results, we explicitly give a method to compute polynomials generating the defining ideal of the Segre locus, for a variety embedded in projective space, in arbitrary characteristic.