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Slope inequalities for fibred surfaces via GIT
Full text linkIn this paper we present a generalisation of a theorem due to Cornalba and Harris, which is an application of Geometric Invariant Theory to the study of invariants of fibrations. In particular, our generalisation makes it possible to treat the problem of bounding the invariants of general fibred surfaces. As a first application, we give a new proof of the slope inequality and of a bound for the invariants associated to double cover fibrations
Fibrations of Campana general type on surfaces
Full text linkWe construct complex surfaces with genus two fibrations over P^1 having special fibres such that the minimum of the multiplicities of the components is ≥ 2 whereas the g.c.d is 1. We can then produce new examples of fibred surfaces without multiple fibres which are of “general type” according to the definition of Campana. We prove that these surfaces are of general type and simply connected; and we compute in some cases their invariants. Moreover, we extend the construction obtaining general type fibrations of any even genus on simply connected surfaces. All our examples are defined over number field
A sharp bound for the slope of double cover fibrations
No full textLet be a fibration of genus whose general fiber is a double cover of a smooth curve of genus . We show that is a sharp lower bound for the slope of when , proving a conjecture of Barja. Moreover, we give a characterization of the fibered surfaces that reach the bound. In the case we obtain the same sharp bound under the additional assumption that the involutions on the general fibers glue to a global involution on
Surfaces on the Severi line
Full text linkLet S be a minimal complex surface of general type and of maximal Albanese dimension; by the Severi inequality one has KS2 ≥ 4χ(OS). We prove that the equality KS2 = 4χ(OS) holds if and only if q(S) := h1(OS) = 2 and the canonical model of S is a double cover of the Albanese surface branched on an ample divisor with at most negligible singularities
The slope of fibred surfaces: unitary rank and Clifford index
Full text linkWe prove new slope inequalities for relatively minimal fibred surfaces,
showing an influence of the relative irregularity, of the unitary rank and of
the Clifford index on the slope. The argument uses Xiao's method and a new
Clifford-type inequality for subcanonical systems on non-hyperelliptic curves.Comment: 23 pages, references added, final version. With respect to the
published version, in the last section there is an added explanation for the
computation of the gonality of the example
The Continuous Rank Function for Varieties of Maximal Albanese Dimension and Its Applications
No full textSlopes of trigonal fibred surfaces and of higher dimensional fibrations
Full text linkWe give lower bounds for the slope of higher dimensional fibrations over
curves under conditions of GIT-semistability of the fibres, using a
generalization of a method of Cornalba and Harris. With the same method we
establish a sharp lower bound for the slope of trigonal fibrations of even
genus and general Maroni invariant; in particular this result proves a
conjecture due to Harris and Stankova-Frenkel.Comment: 11 page
On the complexity group of stable curves
No full textIn this paper, we study combinatorial properties of stable curves. To the dual graph of any nodal curve, there is naturally associated a group, which is the group of components of the Néron model of the generalized Jacobian of the curve. We study the order of this group, called the complexity. In particular, we provide a partial characterization of the stable curves having maximal complexity, and we provide an upper bound, depending only on the genus g of the curve, on the maximal complexity of stable curves; this bound is asymptotically sharp for g ≫ 0. Eventually, we state some conjectures on the behavior of stable curves with maximal complexity, and prove partial results in this direction. © de Gruyter 2011
Linear stability of projected canonical curves, with applications to the slope of fibred surfaces
Full text linkLet f : S → B be a non locally trivial relatively minimal fibred surface. We prove a lower bound for the slope of f depending increasingly from the relative irregularity of f and the Clifford index of the general fibres
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