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On the existence of certain families of curves. (English) Zbl 0541.14024
Let $$V$$ denote an open irreducible subset of the Hilbert-scheme of $$\mathbb P^r$$ parametrizing irreducible and smooth (if $$r\geq 3)$$ resp. nodal (if $$r=2)$$ curves of genus $$g$$ and $$\pi\colon V\to \mathcal M_ g$$ the natural map into the moduli space of curves of genus $$g$$. $$V$$ is called to have the expected number of moduli if $$\dim \pi(V)=\min(3 g-3, 3g - 3+\rho(g,r,n))$$, where $$\rho(g,r,n)$$ is the Brill-Noether number. It is the aim of the paper to construct such families of curves. For plane curves the result is quite complete. It is shown that for all $$g$$ and $$n\geq 5$$ such that $$n-2\leq g\leq \binom{n-1}{2}$$ there is an irreducible component of the family of plane irreducible nodal curves of degree $$n$$ and genus $$g$$, having the expected number of moduli. In the range $$n-2\leq g\leq 3n/2-3$$ (resp. $$2n-4\leq g\leq \binom{n-1}{2}$$ this result was known (resp. independently proved) by P. Griffiths and J. Harris [Duke Math. J. 47, 233–272 (1980; Zbl 0446.14011)] (resp. by Coppens).
For $$r\geq 3$$ the result is: for all $$g$$ and $$n\geq r+1$$ such that $$n-r\leq g\leq(r(n-r)-1)/(r-1)$$ (resp. $$n-3\leq g\leq 3n-18$$ if $$r=3)$$ there is an open set of an irreducible component of the Hilbert scheme of $$\mathbb P^r$$ parametrizing smooth irreducible curves of degree $$n$$ and genus $$g$$, which has the expected number of moduli. An immediate consequence is a special case of a theorem of D. Eisenbud and J. Harris [Invent. Math. 74, 371–418 (1983; Zbl 0527.14022)] on the existence of very ample line bundles of a prescribed type.
The proofs of the theorems are given by induction, the induction step being roughly as follows: Start with a particular curve $$C$$ in $$\mathbb P^n$$ whose existence is known, construct a new curve by adding a particular rational curve $$\gamma$$ and get a new curve $$C'$$ by flat smoothing of $$C\cup \gamma$$.

##### MSC:
 14H10 Families, moduli of curves (algebraic)
##### Keywords:
Hilbert-scheme; moduli space of curves
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##### References:
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