Canonical map of low codimensional subvarieties

Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, low codimensional subvarieties $X$ of $V$ 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 $d$, $K_X$ and $H_X$ denote the degree, the canonical divisor and the general hyperplane section of $X$, $p_g(X^{(h)})$ denotes the geometric genus of the general linear section of $X$ of dimension $h$, and where $\lambda$, $\epsilon_1$ and $\epsilon_2$ are suitable positive real numbers depending only on the dimension of $X$, on $a$ and on the ambient variety $V$. In particular, except for finitely many families of varieties, the canonical map of any irreducible, smooth, projective, low codimensional subvariety $X$ of $V$, 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 $d$ in \PP^3 that is not contained in a surface of degree $<s$. A bound $P(d, s)$ 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 $\mathcal{C}$ of primitive multiple lines and we conjecture that the generic element of $\mathcal{C}$ has good cohomological properties. From the conjecture it would follow that P(d,s)= P_{\textsc{max}}(d,s) for $d=s$ and for every $d \geq 2s-1$. With the aid of \emph{Macaulay2} we checked this holds for $s \leq 120$ 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