Injective linear series of algebraic curves on quadrics
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jective linear series of algebraic curves on quadrics Edoardo Ballico1
· Emanuele Ventura2
Received: 5 May 2020 / Accepted: 11 August 2020 © The Author(s) 2020
Abstract We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic geometry, basic problems still remain unsolved. In this note, we study cuspidal projections of space curves lying on irreducible quadrics (in arbitrary characteristic). Keywords Linear series · Zero-dimensional schemes · Cuspidal projections Mathematics Subject Classification (Primary)14H45 · 14H50; (Secondary)14H20 · 14H51
1 Introduction Projections and singularities of curves are of central importance in algebraic geometry. The projective geometry of singular curves is a delightful chapter of classical algebraic geometry that remains active even up to this date: many questions await to be settled, and in turn they inspire the introduction of tools entailing deformation theory, zerodimensional schemes, and combinatorics, among other techniques. A natural direction of research is the classification of singularities that may arise on a curve X , in some specific ranges of the numerical invariants attached to X . An approach to this classification issue, relying on osculating spaces and combinatorics of semigroups of valuations, has been recently employed in [4]. Other classification results, leveraging the structure of free resolutions of certain ideals, have been achieved in [7].
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Emanuele Ventura [email protected] ; [email protected] Edoardo Ballico [email protected]
1
Università di Trento, 38123 Povo, TN, Italy
2
Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
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ANNALI DELL’UNIVERSITA’ DI FERRARA
In this note, we employ to some extent the standpoint of Greuel, Lossen, and Shustin [11], using the geometry of zero-dimensional schemes and the cohomology of their ideal sheaves, to study cuspidal (or unibranch) singularities. These types of singular points are usually related to some tangency conditions and so carry interesting geometric information about the curve. Let k be an algebraically closed field. Let X be a complete smooth curve of genus g over k, i.e., an integral scheme of dimension one, smooth and proper over k. Every such X is projective and can be embedded in projective 3-space, independently of the characteristic of k. A natural question to wonder about is whether every X admits a projection to P2 with only cuspidal singularities, i.e. X admits a cuspidal projection. Ferrand [9] showed that, when char(k) > 0, if X admits a cuspidal projection then X is a set-theoretic complete intersection. 3 Thereafter, back to characteristic d−1 zero, Piene [24] proved that every X ⊂ P of degree d and genus g, when 2 − g ≤ 3, admits a cuspidal projection. However, a general canonical curve X ⊂ P3 of genus 4 does not admit a cuspidal projec
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