22 research outputs found
Canonical map of low codimensional subvarieties
Fix integers , and . We prove that for certain projective
varieties (e.g. certain possibly singular complete
intersections), there are only finitely many components of the Hilbert scheme
parametrizing irreducible, smooth, projective, low codimensional subvarieties
of such that h^0(X,\Cal O_X(aK_X-bH_X)) \leq \lambda
d^{\epsilon_1}+c(\sum_{1\leq h < \epsilon_2}p_g(X^{(h)})), where ,
and denote the degree, the canonical divisor and the general hyperplane
section of , denotes the geometric genus of the general
linear section of of dimension , and where , and
are suitable positive real numbers depending only on the dimension
of , on and on the ambient variety . In particular, except for
finitely many families of varieties, the canonical map of any irreducible,
smooth, projective, low codimensional subvariety of , is birational.Comment: 31 page
Generically nef vector bundles on ruled surfaces
The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta\u2013Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semista- bility. We establish a Bogomolov-type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section, see Theorem 3.1. This gives an affirmative answer in this case to a problem posed by Peternell [17]. Concerning ruled surfaces with a negative section, we prove a similar result for generically nef vector bundles, with nef and balanced general fiber restriction and with a numerical condition on first Chern class, which is satisfied, for instance, if in its class there is a reduced divisor, see Theorem 3.5. Finally, we use such results to bound the invariants of curve fibrations, which factor through finite covers of ruled surfaces
On the slope inequalities for extremal curves
The present paper concerns the question of the violation of the r-th inequality for extremal curves in Pr , posed in [KM]. We show that the answer is negative in many cases (Theorem 4.13 and Corollary 4.14). The result is obtained by a detailed analysis of the geometry of extremal curves and their canonical model. As a consequence, we show that particular curves on a Hirzebruch surface do not violate the slope inequalities in a certain range (Theorem 6.4)
Generic identifiability of pairs of ternary forms
We prove that two general ternary forms are simultaneously identifiable only
in the classical cases of two quadratic and a cubic and a quadratic form. We
translate the problem into the study of a certain linear system on a projective
bundle on the plane, and we apply techniques from projective and birational
geometry to prove that the associated map is not birational
Equations of tensor eigenschemes
We study schemes of tensor eigenvectors from an algebraic and geometric
viewpoint. We characterize determinantal defining equations of such
eigenschemes via linear equations in their coefficients, both in the general
and in the symmetric case. We give a geometric necessary condition for a
0-dimensional scheme to be an eigenscheme.Comment: 13 page
Eigenschemes of Ternary Tensors
We study projective schemes arising from eigenvectors of tensors, called
eigenschemes. After some general results, we give a birational description of
the variety parametrizing eigenschemes of general ternary symmetric tensors and
we compute its dimension. Moreover, we characterize the locus of triples of
homogeneous polynomials defining the eigenscheme of a ternary symmetric tensor.
Our results allow us to implement algorithms to check whether a given set of
points is the eigenscheme of a symmetric tensor, and to reconstruct the tensor.
Finally, we give a geometric characterization of all reduced zero-dimensional
eigenschemes. The techniques we use rely both on classical and modern complex
projective algebraic geometry.Comment: The title has been slightly modified. We added two algorithms,
testing whether a given configuration of points in the plane is the
eigenscheme of some tensor, and reconstructing a tensor from its eigenpoint
The Maximum Genus Problem for Locally Cohen-Macaulay Space Curves
Let P_{\textsc{max}}(d,s) denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree in \PP^3 that is not contained in a surface of degree . A bound for P_{\textsc{max}}(d,s) has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family of primitive multiple lines and we conjecture that the generic element of has good cohomological properties. From the conjecture it would follow that P(d,s)= P_{\textsc{max}}(d,s) for and for every . With the aid of \emph{Macaulay2} we checked this holds for by verifying our conjecture in the corresponding range
A note on Harris Morrison sweeping families
Harris and Morrison (Invent. Math. 99:321\u2013355, 1990, Theorem 2.5), constructed certain semistable fibrations f:F\u2192Y in k-gonal curves of genus g, such that for every k the corresponding modular curves give a sweeping family in the k-gonal locus (formula presented). Their construction depends on the choice of a smooth curve X. We show that if the genus g(X) is sufficiently high with respect to g, then the ratio (formula presented) is 8 asymptotically with respect to g(X). Moreover, if the conjectured estimates given in Harris and Morrison (Invent. Math. 99:321\u2013355, 1990, pp. 351\u2013352) hold, we show that if g is big enough, then F is a surface of positive index
On the slope of fourgonal semistable fibrations
We bound the slope of sweeping curves in the fourgonal locus of the moduli
space of genus g algebraic curves. Our results follow from some Bogomolov-type
inequalities for weakly positive rank two vector bundles on ruled surfaces.Comment: Change in the title; accepted for publication in Mathematical
Research Letter