Marta Casanellas

STABLE ULRICH BUNDLES

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of generators) on nonsingular cubic surfaces X ⊂ ℙ3. We give necessary and sufficient conditions on the first Chern class D for the existence of stable Ulrich bundles on X of rank r and c1 = D. When such bundles exist, we prove that the corresponding moduli space of stable bundles is smooth and irreducible of dimension D2 - 2r2 + 1 and consists entirely of stable Ulrich bundles (see Theorem 1.1). We are also able to prove the existence of stable Ulrich bundles of any rank on nonsingular cubic threefolds in ℙ4, and we show that the restriction map from bundles on the threefold to bundles on the surface is generically etale and dominant.

Gorenstein liaison and ACM sheaves

Abstract We study Gorenstein liaison of codimension two subschemes of an arithmetically Gorenstein scheme X. Our main result is a criterion for two such subschemes to be in the same Gorenstein liaison class, in terms of the category of ACM sheaves on X. As a consequence we obtain a criterion for X to have the property that every codimension 2 arithmetically Cohen-Macaulay subscheme is in the Gorenstein liaison class of a complete intersection. Using these tools we prove that every arithmetically Gorenstein subscheme of ℙ n is in the Gorenstein liaison class of a complete intersection and we are able to characterize the Gorenstein liaison classes of curves on a nonsingular quadric threefold in ℙ4.

Gorenstein liaison of divisors on standard determinantal schemes and on rational normal scrolls

Abstract Let C⊂ P n be an arithmetically Cohen–Macaulay subscheme. In terms of Gorenstein liaison it is natural to ask whether C is in the Gorenstein liaison class of a complete intersection. In this paper, we study the Gorenstein liaison classes of arithmetically Cohen–Macaulay divisors on standard determinantal schemes and on rational normal scrolls. As main results, we obtain that if C is an arithmetically Cohen–Macaulay divisor on a “general” arithmetically Cohen–Macaulay surface in P 4 or on a rational normal scroll surface S⊂ P n , then C is glicci (i.e. it belongs to the Gorenstein liaison class of a complete intersection).

Invariant Versus Classical Quartet Inference When Evolution is Heterogeneous Across Sites and Lineages.

One reason why classical phylogenetic reconstruction methods fail to correctly infer the underlying topology is because they assume oversimplified models. In this article, we propose a quartet reconstruction method consistent with the most general Markov model of nucleotide substitution, which can also deal with data coming from mixtures on the same topology. Our proposed method uses phylogenetic invariants and provides a system of weights that can be used as input for quartet-based methods. We study its performance on real data and on a wide range of simulated 4-taxon data (both time-homogeneous and nonhomogeneous, with or without among-site rate heterogeneity, and with different branch length settings). We compare it to the classical methods of neighbor-joining (with paralinear distance), maximum likelihood (with different underlying models), and maximum parsimony. Our results show that this method is accurate and robust, has a similar performance to maximum likelihood when data satisfies the assumptions of both methods, and outperform the other methods when these are based on inappropriate substitution models. If alignments are long enough, then it also outperforms other methods when some of its assumptions are violated.

SPIn: Model Selection for Phylogenetic Mixtures via Linear Invariants

In phylogenetic inference, an evolutionary model describes the substitution processes along each edge of a phylogenetic tree. Misspecification of the model has important implications for the analysis of phylogenetic data. Conventionally, however, the selection of a suitable evolutionary model is based on heuristics or relies on the choice of an approximate input tree. We introduce a method for model Selection in Phylogenetics based on linear INvariants (SPIn), which uses recent insights on linear invariants to characterize a model of nucleotide evolution for phylogenetic mixtures on any number of components. Linear invariants are constraints among the joint probabilities of the bases in the operational taxonomic units that hold irrespective of the tree topologies appearing in the mixtures. SPIn therefore requires no input tree and is designed to deal with nonhomogeneous phylogenetic data consisting of multiple sequence alignments showing different patterns of evolution, for example, concatenated genes, exons, and/or introns. Here, we report on the results of the proposed method evaluated on multiple sequence alignments simulated under a variety of single-tree and mixture settings for both continuous- and discrete-time models. In the simulations, SPIn successfully recovers the underlying evolutionary model and is shown to perform better than existing approaches.

The Minimal Resolution Conjecture for Points on the Cubic Surface

In this paper we prove that a generalized version of theMinimal Resolution Conjecture given by Mustaţǎ holds for certain general sets of points on a smooth cubic surface X ⊂ P. The main tool used is Gorenstein liaison theory and, more precisely, the relationship between the free resolutions of two linked schemes. Departament Matematica Aplicada I, ETSEIB UPC., Av. Diagonal 647, 08028-Barcelona. Spain e-mail: marta.casanellas@upc.edu Received by the editors March 4, 2006; revised October 16, 2006. Partially supported by the Ministerio de Educacion y Ciencia of Spain, Programa Ramon y Cajal and MTM2006-E14234-C02-02. AMS subject classification: Primary: 13D02; secondary: 13C40, 14M05, 14M07. c ©Canadian Mathematical Society 2009. 29

The space of phylogenetic mixtures for equivariant models

BackgroundThe selection of an evolutionary model to best fit given molecular data is usually a heuristic choice. In his seminal book, J. Felsenstein suggested that certain linear equations satisfied by the expected probabilities of patterns observed at the leaves of a phylogenetic tree could be used for model selection. It remained an open question, however, whether these equations were sufficient to fully characterize the evolutionary model under consideration.ResultsHere we prove that, for most equivariant models of evolution, the space of distributions satisfying these linear equations coincides with the space of distributions arising from mixtures of trees. In other words, we prove that the evolution of an observed multiple sequence alignment can be modeled by a mixture of phylogenetic trees under an equivariant evolutionary model if and only if the distribution of patterns at its columns satisfies the linear equations mentioned above. Moreover, we provide a set of linearly independent equations defining this space of phylogenetic mixtures for each equivariant model and for any number of taxa. Lastly, we use these results to perform a study of identifiability of phylogenetic mixtures.ConclusionsThe space of phylogenetic mixtures under equivariant models is a linear space that fully characterizes the evolutionary model. We provide an explicit algorithm to obtain the equations defining these spaces for a number of models and taxa. Its implementation has proved to be a powerful tool for model selection.

GenNon-h: Generating multiple sequence alignments on nonhomogeneous phylogenetic trees

BackgroundA number of software packages are available to generate DNA multiple sequence alignments (MSAs) evolved under continuous-time Markov processes on phylogenetic trees. On the other hand, methods of simulating the DNA MSA directly from the transition matrices do not exist. Moreover, existing software restricts to the time-reversible models and it is not optimized to generate nonhomogeneous data (i.e. placing distinct substitution rates at different lineages).ResultsWe present the first package designed to generate MSAs evolving under discrete-time Markov processes on phylogenetic trees, directly from probability substitution matrices. Based on the input model and a phylogenetic tree in the Newick format (with branch lengths measured as the expected number of substitutions per site), the algorithm produces DNA alignments of desired length. GenNon-h is publicly available for download.ConclusionThe software presented here is an efficient tool to generate DNA MSAs on a given phylogenetic tree. GenNon-h provides the user with the nonstationary or nonhomogeneous phylogenetic data that is well suited for testing complex biological hypotheses, exploring the limits of the reconstruction algorithms and their robustness to such models.

EM for phylogenetic topology reconstruction on nonhomogeneous data.

BackgroundThe reconstruction of the phylogenetic tree topology of four taxa is, still nowadays, one of the main challenges in phylogenetics. Its difficulties lie in considering not too restrictive evolutionary models, and correctly dealing with the long-branch attraction problem. The correct reconstruction of 4-taxon trees is crucial for making quartet-based methods work and being able to recover large phylogenies.MethodsWe adapt the well known expectation-maximization algorithm to evolutionary Markov models on phylogenetic 4-taxon trees. We then use this algorithm to estimate the substitution parameters, compute the corresponding likelihood, and to infer the most likely quartet.ResultsIn this paper we consider an expectation-maximization method for maximizing the likelihood of (time nonhomogeneous) evolutionary Markov models on trees. We study its success on reconstructing 4-taxon topologies and its performance as input method in quartet-based phylogenetic reconstruction methods such as QFIT and QuartetSuite. Our results show that the method proposed here outperforms neighbor-joining and the usual (time-homogeneous continuous-time) maximum likelihood methods on 4-leaved trees with among-lineage instantaneous rate heterogeneity, and perform similarly to usual continuous-time maximum-likelihood when data satisfies the assumptions of both methods.ConclusionsThe method presented in this paper is well suited for reconstructing the topology of any number of taxa via quartet-based methods and is highly accurate, specially regarding largely divergent trees and time nonhomogeneous data.

On the Lazarsfeld-Rao property for Gorenstein liaison classes

Abstract We answer a question proposed by Hartshorne about the Lazarsfeld–Rao property for even Gorenstein liaison classes.