Perazzo algebras, strongly Koszulness, Arithmetic Complexes, and Vasconcelos invariant

The following doctorate thesis is based on four different projects. They lie in the intersection between Commutative Algebra, Algebraic Geometry, and Combinatorics. The first project treats Perazzo $3$-folds and the weak Lefschetz property. We characterise the Hilbert function of Perazzo algebras proving that it has a maximum and a minimum. We use this classification to fully display the weak Lefschetz property for those algebras. A classification on minimal Perazzo $3$-folds is given together with a deep study on their geometry and their Jordan type. The second project investigates strongly Koszul algebras. We examine the combinatorial structures underlying these algebras with the aim of describing them in families. Although the results obtained so far are preliminary, they offer new insights into the nature of these algebras. The third regards the study of cohomology groups of line bundles on flag varieties. Using recent results of Raicu and VandeBogert, we study flag varieties on the projective space over the integers extending many known results. A key ingredient is the concept of arithmetic complexes, and a uniform identification formula for these complexes. In the last project, we study the asymptotic properties of the Vasconcelos number. We prove that the (local) Vasconcelos number of $M/I_1^{n_1}\cdots I_r^{n_r}N$ is eventually a minimum of finitely many linear functions on $\underline{n}=(n_1,\dots,n_r)\in\N^r$. In the particular case $r=1$, we prove that the Vasconcelos number is eventually linear and the leading coefficient can be computed using the theory of Rees algebras.