In complex projective 4-space, the only algebraically skew curves are the rational normal curve and the elliptic normal curves; in 3-space, the only one is the twisted cubic.
Terracini loci of curves
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abstract
We study subsets S of curves X whose double structure does not impose independent conditions to a linear series L, but there are divisors D in |L| singular at all points of S. These subsets form the Terracini loci of X. We investigate Terracini loci, with a special look towards their non-emptiness, mainly in the case of canonical curves, and in the case of space curves.
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Algebraically Skew Embeddings of Curves
In complex projective 4-space, the only algebraically skew curves are the rational normal curve and the elliptic normal curves; in 3-space, the only one is the twisted cubic.