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Curves of degree \(r+2\) in \(\mathbb{P}^r\): Cohomological, geometric, and homological aspects. (English) Zbl 1031.14013

It is well-known that the degree and the codimension of a non-degenerate projective variety \(X\) satisfy the inequality: \(\deg (X)\geq \operatorname {codim}(X)+1\). The classification of varieties of minimal degrees has been known for a long time. If \(\deg (X)= \operatorname {codim}(X)+2\), a study of \(X\) from the homological point of view is performed in a paper by Le Tuan Hoa, J. Stückrad and W. Vogel [J. Pure Appl. Algebra 71, 203-231 (1991; Zbl 0749.14032)], but a geometric description has been provided only for smooth curves and surfaces.
In the paper under review, the authors study the curves \(X\) with \(\deg(X)= \operatorname {codim}(X)+3\), contained in \({\mathbb P}^r\) with \(r\geq 3\). The approach is cohomological, i.e. they first classify all possible Hartshorne-Rao modules of these curves: It results that there are four different types, plus an additional type which appears only for \(r=3\). Then they study each type, obtaining the following possibilities for \(X\): It is an embedding of an arbitrary smooth curve of genus 2, or a projection of an elliptic normal curve from a point, or a projection of a rational normal curve from a line. Finally the possible minimal free resolutions of the homogeneous coordinate ring of \(X\) are studied.

MSC:

14H50 Plane and space curves
14H45 Special algebraic curves and curves of low genus
14M07 Low codimension problems in algebraic geometry
14F25 Classical real and complex (co)homology in algebraic geometry

Citations:

Zbl 0749.14032

Software:

Macaulay2
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Full Text: DOI

References:

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