Let $S\subset\mathbb{P}^g$ be a polarised smooth projective K3 surface of genus $g$ (and degree $d=2g-2$) over $\mathbb{C}$. Denote by $\phi: S\to G(3,g+1)$ the Gauss map, taking a point $s\in S$ to its tangent 2-plane $\mathbb{T}_{S,s}\cong\mathbb{P}^2$ in $\mathbb{P}^g$. This is known to always be finite and birational, i.e. it is the normalisation onto its image.

Q: If $g$ is large enough and $S$ is general, is it true that $\phi$ is a closed embedding?

If $g=3$ then the Gauss map $\phi$ of a general quartic K3 $S\subset\mathbb{P}^3$ is ramified along the Hessian curve in $\mathcal{O}_S(8)$, on which it has degree 2, so in particular $\phi$ is not a closed embedding here.

  • $\begingroup$ Can you point me to a reference for the fact that tha Gauss map is finite and birational in the situation described above? $\endgroup$ – Franco Sep 28 '20 at 21:32
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    $\begingroup$ This is true more generally for any smooth projective irreducible variety: the Gauss map of such a thing is finite and birational. See eg Theorem 4.2 in Tevelev's book 'Projective Duality and...' $\endgroup$ – Frank Sep 30 '20 at 8:21

I think that a sufficient condition for the Gauss map to be a closed embedding is that each tangent plane $\mathbb{T}_{S,s}$ intersects the surface $S$ only at the point $s$ and with the expected multiplicity $3$.

In turn, a sufficient condition to guarantee this is that the polarization is $3$-very ample: a paper of Knutsen "On kth-order embeddings of K3 surfaces and Enriques surfaces" characterizes completely the polarizations with this property, and in particular one gets that this happens for a general K3 as soon as $g\geq 7$.

  • $\begingroup$ Hi Frank! You're welcome! $\endgroup$ – Daniele A Dec 13 '17 at 12:44

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