Narrow Results

Let $v_d({\mathbb{P}}^2)\subset |{{𝒪}}_{{\mathbb{P}}^2}(d)|$ denote the $d$-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of $v_d({\mathbb{P}}^2)$ by embedding the blow-up ${\mathrm{\text{bl}}}_{v_d({\mathbb{P}}^2)}|{{𝒪}}_{{\mathbb{P}}^2}(d)|$ into a suitable moduli space of Bridgeland semistable objects on ${\mathbb{P}}^2$.

We prove that if 𝕏 ≔ 𝕏^{(t, s)} is the union of two linear star-configurations of type t × s for 3 ≤ t ≤ 9 and s ≥ t in ℙ², then 𝕏 has generic Hilbert function. We also show that (I_𝕏)_s = {0} when 𝕏 is the union of two linear star-configurations of type s × s for s ≥ 6. Moreover, using the two results with the work in [E. Arrondo and A. Bernardi, On the variety parameterizing completely decomposable polynomials, J. Pure Appl. Algebra215(3) (2011) 201–220], we prove that the secant variety Sec₁(Split_s(ℙⁿ)) is not defective for s ≥ 3 and n ≥ 2.

Let C ⊂ ℙⁿ⁺¹ be a rational normal curve and let X ⊂ ℙⁿ be one of its tangential projection. We describe the X-rank of a point P ∈ ℙⁿ in terms of the schemes evincing the C-rank or the border C-rank of the preimage of P.