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In this note we introduce the notion of $(b,d)$-geprofi sets and study their basic properties. These are sets of $bd$ points in $\mathbb{P}^4$ whose projection from a general point to a hyperplane is a full intersection, i.e., the intersection of a curve of degree $b$ and a surface of degree $d$. We show that such nontrivial sets exist if and only if $b\geq 4$ and $d\geq 2$. Somewhat surprisingly, for infinitely many values of $b$ and $d$ there exist such sets in linear general position. The note contains open questions and problems., Comment: 29 pages

In this paper, we construct an infinite series of line arrangements in characteristic two, each featuring only triple intersection points. This finding challenges the existing conjecture that suggests the existence of only a finite number of such arrangements, regardless of the characteristic. Leveraging the theory of matroids and employing computer algebra software, we rigorously examine the existence and non-existence across various characteristics of line arrangements with up to 19 lines maximizing the number of triple intersection points., Comment: 10 pages

The purpose of this work is to pursue classification of geproci sets. Specifically we classify $[m,n]$-geproci sets which consist of $m=4$ points on each of $n$ skew lines, assuming the skew lines have two transversals in common. We show that in this case $n\leq 6$. Moreover we show that all geproci sets of this type are contained in the \emph{standard construction} for $m=4$ introduced in arXiv:2209.04820. Finally, we propose a conjectural representation for all geproci sets of this type, irrespective of the number $m$ of points on each skew line., Comment: 12 pages, 1 figura

Geproci sets of points in $\mathbb P^3$ are sets whose general projections to $\mathbb P^2$ are complete intersections. The first nontrivial geproci sets came from representation theory, as projectivizations of the root systems $D_4$ and $F_4$. In most currently known cases geproci sets lie on very special unions of skew lines and are known as half grids. For this important class of geproci sets we establish fundamental connections with combinatorics, which we study using methods of algebraic geometry and commutative algebra. As a motivation for studying them, we first prove Theorem A: for a nondegenerate $(a,b)$-geproci set $Z$ with $d$ being the least degree of a space curve $C$ containing $Z$, that if $d\leq b$, then $C$ is a union of skew lines and $Z$ is either a grid or a half grid. We next formulate a combinatorial version of the geproci property for half grids and prove Theorem B: combinatorial half grids are geproci in the case of sets of $a$ points on each of $b$ skew lines when $a\geq b-1\geq 3$. We then introduce a notion of combinatorics for skew lines and apply it to the classification of single orbit combinatorial half grids of $m$ points on each of 4 lines. We apply these results to prove Theorem C, showing, when $n\gg m$, that half grids of $m$ points on $n$ lines with two transversals must be very special geometrically (if they even exist). Moreover, in the case of skew lines having two transversals, our results provide an algorithm for enumerating their projective equivalence classes. We conjecture there are $(m^2-1)/12$ equivalence classes of combinatorial $[m,4]$-half grids in the two transversal case when $m>2$ is prime., Comment: 36 pages

In this short note we develop new methods toward the ultimate goal of classifying geproci sets in $\mathbb P^3$. We apply these methods to show that among sets of $16$ points distributed evenly on $4$ skew lines, up to projective equivalence there are only two distinct geproci sets. We give different geometric distinctions between these sets. The methods we develop here can be applied in a more general set-up; this is the context of the follow-up work arXiv:2308.00761., Comment: 13 pages, to appear in the proceedings of the 2022 Workshop in Cortona: "Lefschetz Properties: Current and New Directions'

We call a set of points $Z\subset{\mathbb P}^{3}_{\mathbb C}$ an $(a,b)$-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point $P$ to a plane is a complete intersection of curves of degrees $a$ and $b$. Examples which we call grids have been known since 2011. The only nongrid nondegenerate examples previously known had $ab=12, 16, 20, 24, 30, 36, 42, 48, 54$ or $60$. Here, for any $4 \leq a \leq b$, we construct nongrid nondegenerate $(a,b)$-geproci sets in a systematic way. We also show that the only such example with $a=3$ is a $(3,4)$-geproci set coming from the $D_4$ root system, and we describe the $D_4$ configuration in detail. We also consider the question of the equivalence (in various senses) of geproci sets, as well as which sets occur over the reals, and which cannot. We identify several additional examples of geproci sets with interesting properties. We also explore the relation between unexpected cones and geproci sets and introduce the notion of $d$-Weddle schemes arising from special projections of finite sets of points. This work initiates the exploration of new perspectives on classical areas of geometry. We formulate and discuss a range of open problems in the final chapter., Comment: 126 pages, 22 figures

Felix Klein in course of his study of the regular icosahedron and its symmetries encountered a highly symmetric configuration of $60$ points in ${\mathbb P}^3$. This configuration has appeared in various guises, perhaps post notably as the configuration of points dual to the $60$ reflection planes in the group $G_{31}$ in the Shephard-Todd list. In the present note we show that the $60$ points exhibit interesting properties relevant from the point of view of two paths of research initiated recently. Firstly, they give rise to two completely different unexpected surfaces of degree $6$. Unexpected hypersurfaces have been introduced by Cook II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated to the configuration of $60$ points is a cone with a single singularity of multiplicity $6$ and the other has three singular points of multiplicities $4,2$ and $2$. Secondly, Chiantini and Migliore observed in 2020 that there are non-trivial sets of points in ${\mathbb P}^3$ with the surprising property that their general projection to ${\mathbb P}^2$ is a complete intersection. They found a family of such sets, which they called grids. An appendix to their paper describes an exotic configuration of $24$ points in ${\mathbb P}^3$ which is not a grid but has the remarkable property that its general projection is a complete intersection. We show that the Klein configuration is also not a grid and it projects to a complete intersections. We identify also its proper subsets, which enjoy the same property. \, Comment: final version, to appear in Michigan. Math. J. Oberwolfach Preprints;2020,19 OWP-2020-19

The main goal of this note is to begin a systematic study on conic-line arrangements in the complex projective plane. We show a de Bruijn-Erd\H{o}s-type inequality and Hirzebruch-type inequality for a certain class of conic-line arrangements having ordinary singularities. We will also study, in detail, certain conic-line arrangements in the context of the geography of log-surfaces and free divisors in the sense of Saito., Comment: 17 pages, 2 figures. Version 3.0, this is the version incorporating referee's remarks

In the present note we give a new proof of a result due to Wiseman and Wilson which establishes an analogue of the Sylvester-Gallai theorem valid for curves of degree two. The main ingredients of the proof come from algebraic geometry. Specifically, we use Cremona transformation of the projective plane and Hirzebruch inequality., Comment: 10 pages, notes after a Lanckorona workshop, October 2014

The Bounded Negativity Conjecture predicts that for any smooth complex surface $X$ there exists a lower bound for the selfintersection of reduced divisors on $X$. This conjecture is open. It is also not known if the existence of such a lower bound is invariant in the birational equivalence class of $X$. In the present note we introduce certain constants $H(X)$ which measure in effect the variance of the lower bounds in the birational equivalence class of $X$. We focus on rational surfaces and relate the value of $H({\mathbb P}^2)$ to certain line arrangements. Our main result is Theorem 3.3 and the main open challenge is Problem 3.10., Comment: v2, rewritten, extra material on arrangements of real lines, to appear in International Mathematics Research Notices

The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such extremal configurations exist. We show that there does not exist a field admitting a configuration of 11 lines with 17 triple points, even though such a configuration is allowed combinatorially. Finally, we present an infinite series of configurations which have a high number of triple intersection points., Comment: 16 pages, 10 figures

The purpose of this note is to study initial sequences of zero-dimensional subschemes of Hirzebruch surfaces and classify subschemes whose initial sequence has the minimal possible growth., Comment: 9 pages