For every choice of degrees, the largest nondegenerate component of the locus where r forms in P^r have positive-dimensional common vanishing is the family of forms all vanishing on a common line.
The moduli space of hypersurfaces whose singular locus has high dimension
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A note on $r$ hypersurfaces intersecting in $\mathbb{P}^r$
For every choice of degrees, the largest nondegenerate component of the locus where r forms in P^r have positive-dimensional common vanishing is the family of forms all vanishing on a common line.