A lower bound for $$\chi ({\mathcal {O}}_S)$$

Let $$(S,{\mathcal {L}})$$ ( S , L ) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $${\mathcal {L}}$$ L of degree $$d > 25$$ d > 25 . In this paper we prove that $$\chi (\mathcal O_S)\ge -\frac{1}{8}d(d-6)$$ χ ( O S ) ≥ - 1 8 d ( d - 6 ) . The bound is sharp, and $$\chi ({\mathcal {O}}_S)=-\frac{1}{8}d(d-6)$$ χ ( O S ) = - 1 8 d ( d - 6 ) if and only if d is even, the linear system $$|H^0(S,{\mathcal {L}})|$$ | H 0 ( S , L ) | embeds S in a smooth rational normal scroll $$T\subset {\mathbb {P}}^5$$ T ⊂ P 5 of dimension 3, and here, as a divisor, S is linearly equivalent to $$\frac{d}{2}Q$$ d 2 Q , where Q is a quadric on T. Moreover, this is equivalent to the fact that a general hyperplane section $$H\in |H^0(S,{\mathcal {L}})|$$ H ∈ | H 0 ( S , L ) | of S is the projection of a curve C contained in the Veronese surface $$V\subseteq {\mathbb {P}}^5$$ V ⊆ P 5 , from a point $$x\in V\backslash C$$ x ∈ V \ C .

Duality, Bridgeland wall-crossing and flips of secant varieties

Let [Formula: see text] denote the [Formula: see text]-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 [Formula: see text] by embedding the blow-up [Formula: see text] into a suitable moduli space of Bridgeland semistable objects on [Formula: see text].

Castelnuovo-Mumford regularity bound for smooth threefolds in ℙ5 and extremal examples

Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙN defined over the complex number field ℂ. X is said to be k-regular if Hi(ℙN, ℐX (k – i)) = 0 for all i ≧ 1, where ℐX is the sheaf of ideals of ℐℙN and Castelnuovo-Mumford regularity reg(X) of X is defined as the least such k. There is a well-known conjecture concerning k-regularity: reg(X) ≦ deg(X) – codim(X) + 1. This regularity conjecture including the classification of borderline examples was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and an optimal bound was also obtained for smooth surfaces (Pinkham, Lazarsfeld). It will be shown here that reg(X) ≦ deg(X) – 1 for smooth threefolds X in ℙ5 and that the only extremal cases are the rational cubic scroll and the complete intersection of two quadrics. Furthermore, every smooth threefold X in ℙ5 is k-normal for all k ≧ deg(X) – 4, which is the optimal bound as the Palatini 3-fold of degree 7 shows. The same bound also holds for smooth regular surfaces in ℙ4 other than for the Veronese surface.