Computable Hilbert Schemes

In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert scheme as subscheme of a suitable Grassmannian and in Chapter 5 we determine a new set of equations of degree lower than the degree of equations known so far. In Chapter 2 we study the most important objects used to project algorithmic techniques, namely Borel-fixed ideals. We determine an algorithm computing all the saturated Borel-fixed ideals with Hilbert polynomial assigned and we investigate their combinatorial properties. In Chapter 3 we show a new type of flat deformations of Borel-fixed ideals which lead us to give a new proof of the connectedness of the Hilbert scheme. In Chapter 4 we construct families of ideals that generalize the notion of family of ideals sharing the same initial ideal with respect to a fixed term ordering. Some of these families correspond to open subsets of the Hilbert scheme and can be used to a local study of the Hilbert scheme. In Chapter 6 we deal with the problem of the connectedness of the Hilbert scheme of locally Cohen-Macaulay curves in the projective 3-space. We show that one of the Hilbert scheme considered a "good" candidate to be non-connected, is instead connected. Moreover there are three appendices that present and explain how to use the implementations of the algorithms proposed.Comment: This is the PhD thesis of the author. Most of the results appeared or are going to appear in some paper. However the thesis contains more detailed explanations, proofs and remarks and it can be used also as handbook for all algorithms proposed and available at http://www.personalweb.unito.it/paolo.lella/HSC/index.html . arXiv admin note: text overlap with arXiv:1101.2866 by other author

Fano 3-folds in codimension 4, Tom and Jerry, Part I

This work is part of the Graded Ring Database project [GRDB], and is a sequel to [Altinok's 1998 PhD thesis] and [Altinok, Brown and Reid, Fano 3-folds, K3 surfaces and graded rings, in SISTAG (Singapore, 2001), Contemp. Math. 314, 2002, pp. 25-53]. We introduce a strategy based on Kustin-Miller unprojection that constructs many hundreds of Gorenstein codimension 4 ideals with 9x16 resolutions (that is, 9 equations and 16 first syzygies). Our two basic games are called Tom and Jerry; the main application is the biregular construction of most of the anticanonically polarised Mori Fano 3-folds of Altinok's thesis. There are 115 cases whose numerical data (in effect, the Hilbert series) allow a Type I projection. In every case, at least one Tom and one Jerry construction works, providing at least two deformation families of quasismooth Fano 3-folds having the same numerics but different topology.Comment: 34pp. This article links to the Graded Ring Database http://grdb.lboro.ac.uk/, and more information is available from webloc. cit. + Downloads. Update includes several clarifications and improvements; results essentially unchanged. To appear in Comp. Mat

Ternary structures in Hilbert spaces

PhDTernary structures in Hilbert spaces arose in the study of in nite dimensional manifolds in di erential geometry. In this thesis, we develop a structure theory of Hilbert ternary algebras and Jordan Hilbert triples which are Hilbert spaces equipped with a ternary product. We obtain several new results on the classi - cation of these structures. Some results have been published in [2]. A Hilbert ternary algebra is a real Hilbert space (V; h ; i) equipped with a ternary product [ ; ; ] satisfying h[a; b; x]; yi = hx; [b; a; y]i for a; b; x and y in V . A Jordan Hilbert triple is a real Hilbert space in which the ternary product f ; ; g is a Jordan triple product. It is called a JH-triple if the identity hfa; b; xg; xi = hx; fb; a; xgi holds in V . JH-triples correspond to a class of Lie algebras which play an important role in symmetric Riemannian manifolds. We begin by proving new structure results on ideals, centralizers and derivations of Hilbert ternary algebras. We characterize primitive tripotents in Hilbert ternary algebras and use them as coordinates to classify abelian Hilbert ternary algebras. We show that they are direct sums of simple ones, and each simple abelian Hilbert ternary algebra is ternary isomorphic to the algebra C2(H;K) of Hilbert-Schmidt operators between real, complex or quaternion Hilbert spaces H and K. Further, we describe completely the ternary isomorphisms and ternary antiisomorphisms between abelian Hilbert ternary algebras. We show that each ternary isomorphism between simple algebras C2(H;K) and C2(H0;K0) is of the form (x) = Jxj where j : H0 ! H and J : K ! K0 are isometries. A ternary anti-isomorphism is of the form (x) = Jx j where j : H0 ! K and J : H ! K0 are isometries. The structures of Hilbert ternary algebras and JH-triples are closely related. Indeed, we show that each JH-triple (V; f ; ; g) admits a decomposi- 6 tion V = Vs L V ? s where (Vs; f ; ; g) is a Hilbert ternary algebra which is usually nonabelian and unless V = Vs, the orthogonal complement V ? s is always a nonabelian Hilbert ternary algebra in the derived ternary product [a; b; c]d = fa; b; cg fb; a; cg. Hence JH-triples provide important examples of nonabelian Hilbert ternary algebras. We determine exactly when Vs and V ? s are Jordan triple ideals of V . We show, in each dimension at least 2, there is a JH-triple (V; f ; ; g) for which V 6= Vs, equivalently, (V; f ; ; g) is not a Hilbert ternary algebra.

Constraints on Macroscopic Realism Without Assuming Non-Invasive Measurability

Macroscopic realism is the thesis that macroscopically observable properties must always have definite values. The idea was introduced by Leggett and Garg (1985), who wished to show a conflict with the predictions of quantum theory. However, their analysis required not just the assumption of macroscopic realism per se, but also that the observable properties could be measured non-invasively. In recent years there has been increasing interest in experimental tests of the violation of the Leggett-Garg inequality, but it has remained a matter of controversy whether this second assumption is a reasonable requirement for a macroscopic realist view of quantum theory. In a recent critical assessment Maroney and Timpson (2017) identified three different categories of macroscopic realism, and argued that only the simplest category could be ruled out by Leggett-Garg inequality violations. Allen, Maroney, and Gogioso (2016) then showed that the second of these approaches was also incompatible with quantum theory in Hilbert spaces of dimension 4 or higher. However, we show that the distinction introduced by Maroney and Timpson between the second and third approaches is not noise tolerant, so unfortunately Allen's result, as given, is not directly empirically testable. In this paper we replace Maroney and Timpson's three categories with a parameterization of macroscopic realist models, which can be related to experimental observations in a noise tolerant way, and recover the original definitions in the noise-free limit. We show how this parameterization can be used to experimentally rule out classes of macroscopic realism in Hilbert spaces of dimension 3 or higher, including the category tested by the Leggett-Garg inequality, without any use of the non-invasive measurability assumption.Comment: 20 pages, 10 figure

Metaphysics of Quantity and the Limit of Phenomenal Concepts

Quantities like mass and temperature are properties that come in degrees. And those degrees (e.g. 5 kg) are properties that are called the magnitudes of the quantities. Some philosophers (e.g., Byrne 2003; Byrne & Hilbert 2003; Schroer 2010) talk about magnitudes of phenomenal qualities as if some of our phenomenal qualities are quantities. The goal of this essay is to explore the anti-physicalist implication of this apparently innocent way of conceptualizing phenomenal quantities. I will first argue for a metaphysical thesis about the nature of magnitudes based on Yablo’s proportionality requirement of causation. Then, I will show that, if some phenomenal qualities are indeed quantities, there can be no demonstrative concepts about some of our phenomenal feelings. That presents a significant restriction on the way physicalists can account for the epistemic gap between the phenomenal and the physical. I’ll illustrate the restriction by showing how that rules out a popular physicalist response to the Knowledge Argument

Discerning Elementary Particles

We extend the quantum-mechanical results of Muller & Saunders (2008) establishing the weak discernibility of an arbitrary number of similar fermions in finite-dimensional Hilbert-spaces in two ways: (a) from fermions to bosons for all finite-dimensional Hilbert-spaces; and (b) from finite-dimensional to infinite-dimensional Hilbert-spaces for all elementary particles. In both cases this is performed using operators whose physical significance is beyond doubt.This confutes the currently dominant view that (A) the quantum-mechanical description of similar particles conflicts with Leibniz's Principle of the Identity of Indiscernibles (PII); and that (B) the only way to save PII is by adopting some pre-Kantian metaphysical notion such as Scotusian haecceittas or Adamsian primitive thisness. We take sides with Muller & Saunders (2008) against this currently dominant view, which has been expounded and defended by, among others, Schr\"odinger, Margenau, Cortes, Dalla Chiara, Di Francia, Redhead, French, Teller, Butterfield, Mittelstaedt, Giuntini, Castellani, Krause and Huggett.Comment: Final Version. To appear in Philosophy of Science, July 200