math.AG — Algebraic Geometry

Ideal generation for general reduced projective points

Contributed by Vlado Vojdanovski

We prove the Ideal Generation Conjecture for general reduced points in projective space over an algebraically closed field of characteristic zero. Equivalently, if the ideal of the points has initial degree D, multiplication by linear forms from degree D to degree D + 1 has maximal rank. The proof combines hyperplane specialization with a construction using two transverse hyperplanes. In the latter construction, motion of their common trace gives the required rank. A uniform induction for all degrees D ≥ 4 reduces the theorem to explicit quadratic and cubic constructions.