The Hilbert curve of a $$4$$-dimensional scroll with a divisorial fiber.(English)Zbl 1436.14018

Let $$(X,L)$$ be a polarized manifold of dimension $$n$$ and let $$Y$$ be a variety of smaller dimension $$m$$. We say that $$(X,L)$$ is a classical scroll if $$X = \mathbb{P}(\mathcal{E})$$ for an ample vector bundle $$\mathcal{E}$$ on $$Y$$, $$L$$ being the tautological line bundle. There is also another notion of a scroll, we say that $$(X,L)$$ is an adjunction theoretic scroll over $$Y$$ if there exists a surjective morphism $$\phi: X \rightarrow Y$$ such that $$K_{X} + (n-m+1)L = \phi^{*}A$$ for some ample line bundle $$A$$ on $$Y$$. In the paper under review, the authors study a $$4$$-dimensional scroll $$(X,L)$$ over $$\mathbb{P}^{3}_{\mathbb{C}}$$ and they provide the equation of its Hilbert curve. Before we present the main construction, let us recall what a Hilber curve is. Let $$(X,L)$$ be a complex polarized manifold of dimension $$n \geq 2$$ and consider $$N(X) := \mathrm{ Num}(X) \otimes_{\mathbb{Z}} \mathbb{C}$$. If $$\mathrm{ rk} \langle K_{X}, L \rangle = 2$$, we can consider the plane $$\mathbb{A}^{2} \subset N(X)$$ generated by the classes $$K_{X}$$ and $$L$$. For any bundle $$D$$ on $$X$$, we consider the complexified polynomial $$p$$ of the Euler-Poincaré characteristic $$\chi(D)$$, where we set $$D = x K_{X} + y L$$ with $$x,y$$ complex numbers, namely $$p(x,y) = \chi(xK_{X} + yL)$$. The Hilbert curve is the complex affine plane curve $$\Gamma$$ of degree $$n$$ defined by $$p(x,y)=0$$. Using symmetry (via Serre duality), we can represent $$\Gamma$$ in the following affine coordinates $$u = x - 1/2$$ and $$v = y$$. Writing our divisor $$D$$ as $$D = \frac{1}{2}K_{X} + \triangle$$ and $$\triangle = uK_{X} + vL$$, the curve $$\Gamma$$ can be represented by $$p(1/2 + u,v) = 0$$.
Set $$X = \mathbb{P}_{Y}(\mathcal{F})$$ and let $$p : X \rightarrow Y$$ be the projection, where $$Y$$ is the blow up of $$\mathbb{P}^{3}_{\mathbb{C}}$$ at a point $$w$$, $$\mathcal{F} = H^{\oplus 2}$$, and $$H = \sigma^{*}\mathcal{O}_{\mathbb{P}^{3}_{\mathbb{C}}}(3) - E$$, where $$\sigma : Y \rightarrow \mathbb{P}^{3}_{\mathbb{C}}$$ is the blow up at $$w$$ and $$E$$ is the exceptional divisior. Denote by $$L$$ the tautological line bundle of $$\mathcal{F}$$ on $$X$$. Observe that $$(X,L)$$ is a (classical) scroll over $$Y$$ via $$p$$.
Main Result. Let $$(X,L)$$ be constructed as above, then the canonical equation of $$\Gamma$$ in the coordinates $$u,v$$ is $p\bigg(\frac{1}{2}+u,v\bigg) = \frac{1}{3}(2u-v)(u-v)(28u^{2} - 38 uv + 13v^{2} - 1) = 0.$

MSC:

 14C20 Divisors, linear systems, invertible sheaves 14N30 Adjunction problems 14J35 $$4$$-folds 14M99 Special varieties

Keywords:

Hilbert curve; scroll; divisorial fiber
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References:

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