Bounding non-rationality of divisors on 3-fold Fano fibrations

In this paper we investigate non-rationality of divisors on 3-fold log Fano fibrations ( X , B ) → Z {(X,B)\to Z} under mild conditions. We show that if D is a component of B with coefficient ≥ t > 0 {\geq t>0} which is contracted to a point on Z, then D is birational to ℙ 1 × C {\mathbb{P}^{1}\times C} , where C is a smooth projective curve with gonality bounded depending only on t. Moreover, if t > 1 2 {t>\frac{1}{2}} , then genus of C is bounded depending only on t.

Characterizing some polarized Fano fibrations via Hilbert curves

The Hilbert curve of a complex polarized manifold [Formula: see text] is the complex affine plane curve of degree [Formula: see text] defined by the Hilbert-like polynomial [Formula: see text], where [Formula: see text] is the canonical bundle of [Formula: see text] and [Formula: see text] and [Formula: see text] are regarded as complex variables. A natural expectation is that this curve encodes several properties of the pair [Formula: see text]. In particular, the existence of a fibration of [Formula: see text] over a variety of smaller dimension induced by a suitable adjoint bundle to [Formula: see text] translates into the fact that the Hilbert curve has a quite special shape. Along this line, Hilbert curves of special varieties like Fano manifolds with low coindex, as well as fibrations over low-dimensional varieties having such a manifold as general fiber, endowed with appropriate polarizations, are investigated. In particular, several polarized manifolds relevant for adjunction theory are completely characterized in terms of their Hilbert curves.

Relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano fibrations

We study the group of relative Fourier–Mukai transforms for Weierstraß fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstraß and Fano or anti-Fano fibrations we describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier–Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determine the group of relative Fourier–Mukai transforms and we prove that the number of relative Fourier–Mukai partners of a given abelian scheme over a normal base is finite.