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Let $H \subset {\mathbb P}^n$ be a real-analytic subvariety of codimensionone induced by a real-analytic curve in the Grassmannian $G(n+1,n)$. Assuming$H$ has a global defining function, we prove $H$ is Levi-flat, the closure ofits smooth points of top dimension is a union of complex hyperplanes, and itssingular set is either of dimension $2n-2$ or dimension $2n-4$. If the singularset is of dimension $2n-4$, then we show the hypersurface is algebraic and theLevi-foliation extends to a singular holomorphic foliation of ${\mathbb P}^n$with a meromorphic (rational of degree 1) first integral. In this case, $H$ isin some sense simply a complex cone over an algebraic curve in ${\mathbb P}^1$.Similarly if $H$ has a degenerate singularity, then $H$ is also algebraic. Ifthe dimension of the singular set is $2n-2$ and is nondegenerate, we show byconstruction that the hypersurface need not be algebraic nor semialgebraic. Weconstruct a Levi-flat real-analytic subvariety in ${\mathbb P}^2$ of realcodimension 1 with compact leaves that is not contained in any properreal-algebraic subvariety of ${\mathbb P}^2$. Therefore a straightforwardanalogue of Chow's theorem for Levi-flat hypersurfaces does not hold.

Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian