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Łucja Farnik

Publications and source records attributed to Łucja Farnik.

13 recordsLinked to original sources
arXiv2026

Enumerative geometry of skew lines in $\mathbb P^3$ with a given associated finite group

For any finite set $\mathcal L$ of 3 or more skew lines in $\mathbb P^3_{\overline{K}}$ over an algebraically closed field $\overline{K}$ of arbitrary characteristic, there is a canonical associated subgroup $G_{\mathcal L}$ of ${\rm PGL}_2(\overline{K})$. Given a finite subgroup $G\subset{\rm PGL}_2(\overline{K})$ we study which configurations of lines have $G_{\mathcal L}=G$. We derive an upper bound on the number $|\mathcal L|$ of lines in terms of the order $|G|$ of the group $G$ and as an application we classify up to projective equivalence which sets $\mathcal L$ in $\mathbb P^3_{\mathbb C}$ have $G_{\mathcal L}=G$ for certain finite nonabelian groups $G$.

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arXiv2026

Weddle schemes

The classical Weddle surface is the locus of vertices of quadric cones through six points in $\mathbb{P}^3$ in linear general position. Equivalently, it is the closure of the locus of centers of projection from which those six points map to six points on a plane conic. Motivated by this 1850 construction of T. Weddle, we introduce $d$-Weddle schemes for finite point sets $Z\subset \mathbb{P}^n$, defined by an analogous projection-to-degree-$d$ condition. Our main tool is Macaulay duality, which yields a natural multiplication map in an Artinian algebra defined by powers of linear forms. This viewpoint connects $d$-Weddle schemes to unexpected cones and interprets them as non-Lefschetz loci for these multiplication maps. Parallel to this, we give an analysis from the point of view of interpolation matrices, and we explain the connections between these approaches. For a general set $Z\subset \mathbb{P}^n$ of $\binom{d+n}{n}$ points, we show that the $d$-Weddle scheme is a hypersurface and we compute its degree. We also study general sets whose cardinalities are "near" such a binomial coefficient, where the Weddle scheme has higher codimension. Returning to sets of six points (not always in linear general position), we discuss special configurations in which the appropriate Weddle scheme is reducible, or even nonreduced.

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arXiv2026

Intersection of curves in projective 4 space

Given two distinct reduced, irreducible curves of given degrees, contained in projective space but whose union is not contained in a hyperplane, what is the largest number of points of intersection they can have? When the projective space is the plane, this is trivial. For projective 3 space this problem was solved independently by Diaz and by Giuffrida in 1986. They showed that two curves achieving the maximum number of intersection points have to be rational curves on a smooth surface of minimal degree, i.e., a quadric surface. Note that these curves are far from being arithmetically Cohen-Macaulay. In contrast, Hartshorne and Miró-Roig addressed this problem in 2015 for space curves under the assumption that the curves are arithmetically Cohen-Macaulay (ACM), introducing very deep techniques and obtaining very different results from Diaz and Giuffrida. Diaz and Giuffrida also gave initial results in dimensions greater than 3. Here we continue this study for dimension 4. We introduce a number B defined in terms of the degrees of the curves and prove that when both curves lie on a surface of minimal degree (thus a cubic surface) then the number of points of intersection is at most B. Moreover, we conjecture that B is always an upper bound and we prove this conjecture in many cases, including when at least one of the curves is ACM. Our approach focuses on the genera of the curves and their union. In addition we define a second number B' in terms of the degrees and the genus of the union which we can show bounds the number of points of intersection above, and we use a variety of methods to study how B and B' compare.

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arXiv2025

Combinatorics of skew lines in $\mathbb P^3$ with an application to algebraic geometry

This article introduces a previously unrecognized combinatorial structure underlying configurations of skew lines in $\mathbb{P}^3$, and reveals its deep and surprising connection to the algebro-geometric concept of geproci sets. Given any field $\mathbb{K}$ and a finite set $\mathcal L$ of 3 or more skew lines in $\mathbb{P}^3_\mathbb{K}$, we associate to it a group $G_{\mathcal L}$ and a groupoid $C_{\mathcal L}$ whose action on the union $\cup_{L\in\mathcal L}L$ provides orbits which have a rich combinatorial structure. We characterize when $G_{\mathcal L}$ is abelian and give partial results on its finiteness. The notion of \emph{collinearly complete} subsets is introduced and shown to correspond exactly to unions of groupoid orbits. In the case where $\mathbb{K}$ is a finite field and $\mathcal L$ is a full spread in $\mathbb{P}^3_\mathbb{K}$ (i.e., every point of $\mathbb{P}^3_\mathbb{K}$ lies on a line in $\mathcal{L}$), we prove that $G_{\mathcal L}$ being abelian characterizes the classical spread given by the fibers of the Hopf fibration. Over any algebraically closed field, we establish that finite unions of $C_{\mathcal L}$-orbits are geproci sets - that is, finite sets whose general projections to a plane are complete intersections. Furthermore, we prove a converse: if $\mathbb{K}$ is algebraically closed and $Z \subset \mathbb{P}^3_\mathbb{K}$ is a geproci set consisting of $m$ points on each of $s \geq 3$ skew lines $\mathcal L$ where the general projection of $Z$ is a complete intersection of type $(m, s)$, then $Z$ is a finite union of orbits of $C_{\mathcal L}$. This work thus uncovers a profound combinatorial framework governing geproci sets, providing a new bridge between incidence combinatorics and algebraic geometry.

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arXiv2024

Cactus varieties of sufficiently ample embeddings of projective schemes have determinantal equations

For a fixed projective scheme X, a property P of line bundles is satisfied by sufficiently ample line bundles if there exists a line bundle L_0 on X such that P(L) holds for any L with (L - L_0) ample. As an example, sufficiently ample line bundles are very ample, moreover, for a normal variety X, the embedding corresponding to sufficiently ample line bundle is projectively normal. The grandfather of such properties and a basic ingredient used to study this concept is Fujita vanishing theorem, which is a strengthening of Serre vanishing to sufficiently ample line bundles. The r-th cactus variety of X is an analogue of secant variety and it is defined using linear spans of finite schemes of degree r. In this article we show that cactus varieties of sufficiently ample embeddings of X are set-theoretically defined by minors of matrices with linear entries. The topic is closely related to conjectures of Eisenbud-Koh-Stillman, which was proved by Ginensky in the case X a smooth curve. On the other hand Sidman-Smith proved that the ideal of sufficiently ample embedding of any projective scheme X is generated by 2 x 2 minors of a matrix with linear entries.

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arXiv2024

Finite sets of points in $\mathbb{P}^4$ with special projection properties

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.

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arXiv2022

Configurations of points in projective space and their projections

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.

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arXiv2020

Seshadri constants on abelian and bielliptic surfaces -- potential values and lower bounds

In this note we contribute to the study of Seshadri constants on abelian and bielliptic surfaces. We specifically focus on bounds that hold on all such surfaces, depending only on the self-intersection of the ample line bundle under consideration. Our result improves previous bounds and it provides rational numbers as bounds, which are potential Seshadri constants.

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arXiv2020

Rationality of Seshadri constants on general blow ups of $\mathbb{P}^2$

Let $X$ be a projective surface and let $L$ be an ample line bundle on $X$. The global Seshadri constant $\varepsilon(L)$ of $L$ is defined as the infimum of Seshadri constants $\varepsilon(L,x)$ as $x\in X$ varies. It is an interesting question to ask if $\varepsilon(L)$ is a rational number for any pair $(X, L)$. We study this question when $X$ is a blow up of $\mathbb{P}^2$ at $r \ge 0$ very general points and $L$ is an ample line bundle on $X$. For each $r$ we define a $\textit{submaximality threshold}$ which governs the rationality or irrationality of $\varepsilon(L)$. We state a conjecture which strengthens the SHGH Conjecture and assuming that this conjecture is true we determine the submaximality threshold.

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arXiv2019

On the unique unexpected quartic in $\mathbb{P}^2$

The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre-Harbourne-Gimigliano-Hirschowitz conjecture. When we consider fixed points in special position, even more interesting situations may occur. Recently Di Gennaro, Ilardi and Vallès discovered a special configuration $Z$ of nine points with a remarkable property: a general triple point always fails to impose independent conditions on the ideal of $Z$ in degree four. The peculiar structure and properties of this kind of \textit{unexpected curves} were studied by Cook II, Harbourne, Migliore and Nagel. By using both explicit geometric constructions and more abstract algebraic arguments, we classify low degree unexpected curves. In particular, we prove that the aforementioned configuration $Z$ is the unique one giving rise to an unexpected quartic.

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arXiv2018

On the parameter space of Böröczky configurations

Böröczky configurations of lines have been recently considered in connection with the problem of the containment between symbolic and ordinary powers of ideals. Here we describe parameter families of Böröczky configurations of 13, 14, 16, 18 and 24 lines and investigate rational points of these parameter spaces.

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arXiv2017

Containment problem and combinatorics

In this note we consider two configurations of twelve lines with nineteen triple points (i.e., points where three lines meet). Both of them have the same combinatorial features. In both configurations nine of twelve lines have five triple points and one double point, and the remaining three lines have four triple points and three double points. Taking the ideal of the triple points of these configurations we discover that, quite surprisingly, for one of the configurations the containment $I^{(3)} \subset I^2$ holds, while for the other it does not. Hence for ideals of points defined by configurations of lines the (non)containment of a symbolic power in an ordinary power is not determined alone by combinatorial features of the arrangement. Moreover, for the configuration with the non-containment $I^{(3)} \nsubseteq I^2$ we examine its parameter space, which turns out to be a rational curve, and thus establish the existence of a rational non-containment configuration of points. Such rational examples are very rare.

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