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I can't understand the proof of Lemma3.3 in Stability of genus 5 canonical curves.

Let $C$ be a complete intersection of three quadrics in $\mathbb{P}^4$ and let $\Lambda$ be the net of quadrics containing $C$. If the discriminant $\Delta(C)$ is reduced plane curve, then $C$ lies on a smooth quartic del Pezzo surface, defined by any pencil $l\subset \Lambda$ transverse to $\Delta(C)$. Why?

Conversely, if $C$ lies on a smooth quartic del Pezzo $P$ in $\mathbb{P}^4$, then the pencil of quadrics containing $P$ has a reduced discriminanat. Why?

Does only smooth quartic del Pezzo surface(among quartic surfaces in $\mathbb{P}^4$) have a reduced discriminant?

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  • $\begingroup$ Could you please say exactly what you mean by the discriminant of the curve C ? There is a lot of information on this set up scattered in the literature. I have yet to have found proofs for almost any of the facts I've found. $\endgroup$
    – meh
    Commented Sep 22, 2014 at 2:38
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    $\begingroup$ @aginensky: This is the discriminant of the corresponding net of quadrics. It is a curve in $\mathbb{P}^2$, which parametrises the singular quadrics in the net. $\endgroup$ Commented Sep 22, 2014 at 9:03

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If $\Delta (C)$ is reduced, a general line $\ell\cong\mathbb{P}^1$ intersects it in 5 distinct points $\lambda _1,\ldots ,\lambda _5$. This implies that the corresponding pencil of quadrics can be written in an appropriate system of coordinates $\alpha (\sum x_i^2)+\beta (\sum \lambda _ix_i^2)=0$ (see e.g. Dolgachev's book). In particular its base locus is a smooth quartic surface.

Conversely, if $C$ lies in a smooth quartic del Pezzo surface, the corresponding line meets $\Delta (C)$ in 5 distinct points; this implies that $\Delta (C)$ is reduced.

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