Existence and reducibility of the Hilbert scheme of smooth and linearly normal curves in \(\mathbb{P}^r\) of relatively high degrees. (English) Zbl 1497.14008
This paper is one in a long list of papers by several specialists concerning the following question, called the Modified Severi Conjecture. Fix integers \(g\ge 0\), \(r\ge 3\), \(d\ge r\) such that \((g,r,d)\) is in the Brill-Noether range \(\rho(g,r,d)\ge 0\). Let \(H^L_{d,g,r}\) denote the Hilbert scheme of all smooth and linearly normal curves \(C\subset \mathbb{P}^r\) of degree \(d\) and genus \(g\). Is \(H^L_{d,g,r}\) non-empty? Is it irreducible? Has it the expected dimension? The restriction to components whose general member is linearly normal is essential. The paper gives an answer in several cases, e.g. to all 3 question under for some \(r, d, g\). For instance if \(d=g+r-2\) if \(g\le r+3\) all answers are Yes, while \(H^L_{g-r+2,g,r} =\emptyset\) if \(g\le r+2\). For \(d=g+r-3\) the paper proves several existence and irreducibility statements. The paper also lists the results previously known and so it may be used as an up to date survey.
Reviewer: Edoardo Ballico (Povo)
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