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The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System $S(t,n,v)$ we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement.
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Given a homogeneous ideal $I subseteq k[x_0,dots,x_n]$, the Containment problem studies the relation between symbolic and regular powers of $I$, that is, it asks for which pair $m, r in mathbb{N}$, $I^{(m)} subseteq I^r$ holds. In the last years, several conjectures have been posed on this problem, creating an active area of current interests and ongoing investigations. In this paper, we investigated the Stable Harbourne Conjecture and the Stable Harbourne -- Huneke Conjecture and we show that they hold for the defining ideal of a Complement of a Steiner configuration of points in $mathbb{P}^{n}_{k}$. We can also show that the ideal of a Complement of a Steiner Configuration of points has expected resurgence, that is, its resurgence is strictly less than its big height, and it also satisfies Chudnovsky and Demaillys Conjectures. Moreover, given a hypergraph $H$, we also study the relation between its colourability and the failure of the containment problem for the cover ideal associated to $H$. We apply these results in the case that $H$ is a Steiner System.
We present a direct and fairly simple proof of the following incidence bound: Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in ${mathbb R}^d$, for $dge 3$, which lie in a common algebraic two-dimensional surface of degree $D$ that does not contain any 2-flat, so that no 2-flat contains more than $s le D$ lines of $L$. Then the number of incidences between $P$ and $L$ is $$ I(P,L)=Oleft(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}min{n,D^{2}}^{1/3}s^{1/3} + m + nright). $$ When $d=3$, this improves the bound of Guth and Katz~cite{GK2} for this special case, when $D$ is not too large. A supplementary feature of this work is a review, with detailed proofs, of several basic (and folklore) properties of ruled surfaces in three dimensions.
In this work we study $k$-type uniform Steiner bundles, being $k$ the lowest degree of the splitting. We prove sharp upper and lower bounds for the rank in the case $k=1$ and moreover we give families of examples for every allowed possible rank and explain which relation exists between the families. After dealing with the case $k$ in general, we conjecture that every $k$-type uniform Steiner bundle is obtained through the proposed construction technique.
A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.
This is a report on a joint project in experimental mathematics with Jonas Bergstrom and Carel Faber where we obtain information about modular forms by counting curves over finite fields.
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