Injective linear series of algebraic curves on quadrics

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).

Algebraic compressed sensing

“…We conclude the discussion of identifiability emphasizing the intricacy of the questions in this section, even in the mathematically friendlier setting of complex projective geometry. For example, it is an open problem whether a d-dimensional complex projective variety may be algebraically injected to a projective space of dimension 2d using arbitrary algebraic maps; see [84][85][86] and [87,Question 3]. A weaker negative result is known: there are algebraic curves X in a complex 3-dimensional projective space that cannot be injectively projected to the projective plane [88].…”

Algebraic compressed sensing

“…We conclude the discussion of identifiability emphasizing the intricacy of the questions in this section, even in the mathematically friendlier setting of complex projective geometry. For example, it is an open problem whether a d-dimensional complex projective variety may be algebraically injected to a projective space of dimension 2d using arbitrary algebraic maps; see [7,41,42] and [70, Question 3]. A weaker negative result is known: there are algebraic curves X (d = 1) in a 3dimensional projective space (n = 3) that cannot be injectively projected to the complex projective plane (s = 2) [77].…”