Nagata type statements

Nagata solved Hilbert's 14-th problem in 1958 in the negative. The solution naturally lead him to a tantalizing conjecture that remains widely open after more than half a century of intense efforts. Using Nagata's theorem as starting point, and the conjecture, with its multiple variations, as motivation, we explore the important questions of finite generation for invariant rings, for support semigroups of multigraded algebras, and for Mori cones of divisors on blown up surfaces, and the rationality of Waldschimdt constants. Finally we suggest a connection between the Mori cone of the Zariski-Riemann space and the continuity of the Waldschmidt constant as a function on the space of valuations.Comment: 45 pages. These notes correspond to the course of the same title given by the first author in the workshop "Asymptotic invariants attached to linear series" held in the Pedagogical University of Cracow from May 16 to 20, 201

Geometry of syzygies via Poncelet varieties

We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of V_n=H^0({\P^1},\O_{\P^1}(n)). We define $\frak{X}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on $\P^1$ whose basis realize a fixed number of polynomial relations of fixed degree, say a fixed number of syzygies of a certain degree. The first result of this paper is the computation of the dimension of $\frak{X}_{k,r,d}$. In the second part we make a link between $\frak{X}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the Poncelet varieties

Euler characteristic for teachers

In the present paper we explain the well-known Euler formula for solids, as an interesting argument to be dealt with students, in order to point out that in mathematics, and particularly in geometry, there are many different ways to investigate objects. It is also explained with examples the topological reason for which, in defining the Euler Characteristic of a polyhedral surface in the space, it is necessary to use a triangulation, instead of a general tassellation

Curves with constant width: a proposal for a geometry unit

En este trabajo se propone una actividad de matemática para la educación primaria y una posible extensión a la escuela secundaria, basada en resolución de problemas y debates, empezando con monedas de todo el mundo de formas diferentes de las comunesAn activity for primary school and its extension to high school in geometry is proposed, based on guided classroom discussion and problem solving, starting from unusual shapes of coins worldwid

A probabilistic approach for the estimation of earthquake source parameters from spectral inversion

The characterization of the mechanisms of earthquake generation and propagation is a major challenge in understanding the Earth engine. Although the seismic rupture non-linearly combines several space and time scales, some macroscopic parameters can provide insights in its evolution, such as the earthquake size and the stress drop released during a seismic event. However, the estimation of these parameters is very uncertain (Cotton et al., 2013), owing to uncertainties in data and models and to the strong coupling between source effects and wave propagation up to the observation sites. The objective of this thesis is the characterization of the seismic source parameters using the amplitude spectrum of the displacement records and assuming that the earthquake behaves as a circular crack (Keilis-Borok, 1959). Several methods for the characterization of the source using a spectral analysis have been proposed in literature. Systematic comparison between different methodologies highlighted the dependence of the results on the fitting model, due to the high correlation between the parameters, especially comparing EGF and TGF based techniques (Ide et al. 2003; Oye et al., 2005). A probabilistic approach can allow to investigate such a correlation, defining a probability density function (PDF) in the parameter space and allowing for a consistent estimate of the uncertainties. Using the probabilistic framework developed by Tarantola (2005), and specifically the notion of conjunction of states of information, I developed a probabilistic approach to retrieve the source parameters seismic moment (through the low-frequency spectral level), the corner frequency (that is a proxy of the rupture length) and the high-frequency decay parameter. Information on the source of an earthquake requires the modeling of the wave propagation too; I choose to use in this work a theoretical Green’s function, adding one parameter to invert related to the propagation (a frequency-independent Q-factor) beyond the three source parameter that I want to retrieve. I model the observations with an operator, defined on these four parameters, which is non-linear; thus, a global exploration of the model space is required in order to find the best solution to describe the data. Additionally, the joint a-posteriori probability density function (PDF) is computed around the best model, to extract the correlation matrix of the parameters. This allows to obtain estimates and uncertainties from the PDF, that are taking into account the correlations. The global exploration relies on the building of a Markov chain in the parameter space and on the combination of a deterministic minimization with a random exploration of the space (Basin-Hopping method, Wales and Doye, 1997; Wales, 2003). The main advantages of this new methodology are the following : • A fully probabilistic approach associated with a global exploration method can provide a robust information about the “best-fit” model, with correct estimation of uncertainties and parameter correlation. • The shape of the estimated PDF can assess the quality of the solutions, allowing to rule out noisy data and thus enabling the use of the method for automatic processing of large datasets. I performed three applications of the method. In Chapter 4, I analyzed the Central Italy 2016-2017 sequence, characterizing the source of all the earthquakes with Ml > 4 (56 events); in Chapter 5 I characterized the source of more than 10000 LFEs occurred in the Nankai region (Japan) during the period 2012-2016; in Chapter 6 I analyzed the micro-seismicity (Ml between 0 and 4.5, 1061 events) occurred from 2016 to 2017 in the Northern Ibaraki region (Japan)

On complete intersections containing a linear subspace

Consider the Fano scheme $F_k(Y)$ parameterizing $k$-dimensional linear subspaces contained in a complete intersection $Y \subset \mathbb{P}^m$ of multi-degree $\underline{d} = (d_1, \ldots, d_s)$. It is known that, if $t := \sum_{i=1}^s \binom{d_i +k}{k}-(k+1) (m-k)\leqslant 0$ and $\Pi_{i=1}^sd_i >2$, for $Y$ a general complete intersection as above, then $F_k(Y)$ has dimension $-t$. In this paper we consider the case $t> 0$. Then the locus $W_{\underline{d},k}$ of all complete intersections as above containing a $k$-dimensional linear subspace is irreducible and turns out to have codimension $t$ in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general $[Y]\in W_{\underline{d},k}$ the scheme $F_k(Y)$ is zero-dimensional of length one. This implies that $W_{\underline{d},k}$ is rational.Comment: 6 pages, the collaboration has benefitted of funding from the research project \emph{"Families of curves: their moduli and their related varieties"} (CUP: E81-18000100005) - Mission Sustainability - University of Rome Tor Vergat

On the functioning of business groups: economic rationale, financial structure and performance

Business groups are organizational forms which stand in the middle between hierarchies and markets, in the sense that they formally rely upon within-boundaries mechanisms of coordination while occasionally replicating market-like methods. Business groups (hereinafter also BGs) are collections of independent firms bound together under a unique ownership that gain some advantage from common affiliation. In my dissertation, I explore the role played by business groups in Italy in different periods of time. I describe the main features of Italian business groups (predominantly family- or state-owned) while assessing their relevance within the national industrial landscape. Once determined the effective composition of Italian control pyramids, I try to verify whether (i) internal capital markets do exist and the extent to which they work actively and efficiently; (ii) group affiliation facilitates member firms in terms of performance, capital structure choices and financial outcomes in both good and bad times. The thesis consists of three chapters. The first chapter reviews and systematizes the extant literature on business groups, with the aim of identifying avenues for future research through survey of past scholarship. The second chapter investigates how an exogenous shock affects capital reallocation among firms affiliated to business groups and their subsequent investment decisions. We use unique financial and accounting micro-data which allow us to have a deep knowledge of the ownership links between member firms. Thus, we reconstruct almost all Italian business groups as hierarchies. Our sample period encompasses the global 2008 financial crisis, a natural event which is likely to exacerbate external financial constraints and, consequently, the impact of internal capital markets on funds distribution. In particular, we test whether intra-group transfers provide BG-affiliated firms with a substantial financing advantage when compared to their (similarly constrained) standalone peers. Finally, the third chapter extends evidence collected in the second essay and compares a cohort of affiliated firms with a cohort of comparable independent ones to measure the potential advantage group members may have vis-\ue0-vis independent ones in performance outcomes during times of market turmoil

Innovation and Development. The Evidence from Innovation Surveys.

In this article we investigate the existing evidence on innovation produced by innovation surveys in developing and emerging countries in Europe, Asia, Africa and Latin America. We review the relevant literature, discuss methodological issues, and analyse the results for the countries with the most comparable surveys, considering the well established findings of innovation surveys for Europe as a benchmark. From the evidence we considered, regional patterns are identified and some stylized facts on innovation and development are proposed, pointing out the specificity of innovation processes in economies engaged in industrialisation and catching-up.Innovation Surveys, Patterns of Innovation, Emerging Countries.