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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
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References:

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