Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree d rational curves in $\mathbb{P}^n$ when $d - n \leq 3$ and $d < 2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d < 2n$, the arithmetic genus of any nondegenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d - n$.Comment: 34 pages, 11 tables, 2 figures; v2 added references and made minor corrections; v3 more minor revisions, to appear in IMR

Modeling the 21cm global signal from first stars and black holes.

The aim of this project is to predict the 21-centimeter global signal generated by the transition between two hyperfine levels of the atomic hydrogen coming from the high-redshift universe. This signal is supposed to be generated during the epoch of formation of first structures because, once luminous objects are formed, they will emit UV radiation that penetrates primordial hydrogen and that is able to alter its spin temperature. This process ends up in a global signal in the 21-centimeter band that begins when the first structures start to form and so the spin temperature decouples from the photon temperature. The signal saturates when reionization completes since there is no more atomic hydrogen that can emit or absorb in the 21- centimeter band. The 21 centimeter signal thus, depends on the ionization and thermal histories of the intergalactic medium. Three relevant processes determine these two histories: X-ray heating responsible for the increase of the kinetic temperature of the gas, Lyman-alpha photons that couple the spin temperature of the gas to its kinetic temperature and UV ionizing photons that drive the cosmic reionization. A crucial role is thus played by the sources that firstly formed in the universe (stars and black holes) since they can emit photons at all the different frequencies of our interest. In the first part of the thesis, we adopted a standard analytical model for structure formation based on the Press-Schechter formalism in order to obtain thermal and ionization histories (and thus the 21-centimeter global signal) consistent with already published results. In the second part of the thesis, we used the semi-numerical code Cosmic Archaeology Tool (shortly CAT) developed within our research group (Trinca et al., 2021). CAT is able to follow the evolution of dark matter halos tracking merger history using the extended PressSchechter formalism and provides an ab.initio description of their baryonic evolution, starting from the formation of the first stars and black holes in mini-halos at z=20-30. The model is well anchored to observations of galaxies and AGN at z<6 and it predicts a reionization history consistent with observations. We then estimated the 21- centimeter global signal using the same formalism described above but with the rate of formation and emission properties of the sources (stars and accreting black holes) provided by CAT. We obtained a 21- centimeter global signal with an absorption feature between z=23 and z=19 and with a depth of 150 mK. The timing and the depth of this feature is consistent with many other semi-numerical models that account for star formation in mini-halos, but it is not consistent with the detection claimed by the EDGES collaboration (Bowman et al., 2018) since it has a depth three times stronger. We tried to reproduce this depth considering an additional radio background produced by the emission of early accreting black hole seeds adopting the same formalism of Ewall-Wice et al. (2018). We found that considering only black holes which are accreting with an Eddington ratio larger than 0.01 we may reproduce the observed depth of 500 mK

Minimal degree equations for curves and surfaces (variations on a theme of Halphen)

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such collections of varieties: their minimal degree and their maximal number of linearly independent smallest degree hypersurfaces passing through them. We show results for curves and surfaces, and pose several questions.Comment: 15 p

Injective linear series of algebraic curves on quadrics

We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic geometry, basic problems still remain unsolved. In this note, we study cuspidal projections of space curves lying on irreducible quadrics (in arbitrary characteristic).Comment: Expanded material from previous versio

On the Koiran-Skomra's question about Hessians

We give a negative answer to a question of Koiran and Skomra about Hessians, motivated by Kayal's algorithm for the equivalence problem to the Fermat polynomial. We conjecture that our counterexamples are the only ones. We also study a local version of their question.Comment: 14 p