Search arXiv⌕ Search

arXiv subjects

Tim Ryan

Publications and source records attributed to Tim Ryan.

15 recordsLinked to original sources
arXiv2026

Seeking Help in the Digital Age: A Cross-Platform Analysis of Online Support Systems for Technology-Facilitated Abuse Victims

Technology-facilitated abuse (TFA), the use of digital technologies to stalk, harass, monitor or threaten others, has become a pervasive form of interpersonal harm. As victims turn to online sources for guidance, responses can shape how they assess risks, interpret abuse, and choose protective actions. We present a large-scale evaluation of online support for TFA victims across three channels: web search, peer-support forums, and conversational AI systems. Drawing on a decade of victim narratives from r/Stalking, we use qualitative coding and supervised classifiers to construct a dataset of TFA queries spanning 11 categories of technology misuse. We simulate these queries across the three channels and evaluate responses using a unified framework spanning technical, social, and safety dimensions. The framework assesses relevance, accuracy, actionability, persuasiveness, and understandability, alongside platform risks and support characteristics, including social-engineering risk, toxicity, empathy, bias, risky guidance, and support information. We build and validate automated classifiers to scale the evaluation. Our findings reveal differences in support quality across platforms. Google Search and general-purpose LLMs provide more relevant and actionable guidance than Reddit discussions, yet none consistently provide safe, trauma-informed support. More than 65% of victim queries encounter potentially malicious links in search results, over 20% of Reddit discussions contain toxic responses, and conversational AI systems frequently fail to provide risk-aware guidance or concrete support resources. Surprisingly, domain-specific survivor-support chatbots underperform general-purpose LLMs across most dimensions. These findings expose weaknesses in digital support for TFA victims and highlight the need for safety-centered design, evaluation, and deployment of future support technologies.

cs.CY↗
arXiv2024

Geometry of syzygies of sheaves on $\mathbb{P}^2$ via interpolation and Bridgeland stability

We show that the minimal free resolution of a general semi-stable sheaf $U$ on $\mathbb{P}^2$ contains a subcomplex that determines an extremal ray of the cone of effective divisors of its moduli space. We provide evidence that this is part of a general phenomenon in which minimal free resolutions, for distinct Betti tables, contain subcomplexes depending on wall-crossing. From this viewpoint, we provide new computations of the movable cones and Mori decompositions of some moduli spaces of sheaves using syzygies.

math.AG↗
arXiv2024

Smooth Surfaces with Maximal Lines

We prove that a smooth projective surface of degree $d$ in $\mathbb P^3$ contains at most $d^2(d^2-3d+3)$ lines. We characterize the surfaces containing exactly $d^2(d^2-3d+3)$ lines: these occur only in prime characterize $p$ and, up to choice of projective coordinates, are cut out by equations of the form $x^{p^{e}+1}+y^{p^{e}+1}+z^{p^{e}+1}+ w^{p^{e}+1} = 0.$

math.AG↗
arXiv2022

Maximal skew sets of lines on a Hermitian surface and a modified Bron-Kerbosch algorithm

In this paper, we study maximal sets of skew lines on Hermitian surfaces. We give a new algorithm to compute these sets and give some computational results for Hermitian surfaces of degrees 3,4, and 5. In more generality, this algorithm solves a new variant of the clique listing problem, which may be more approachable than the classical problem. Finally, we explicitly construct a large skew set of lines on Hermitian varieties of any degree and use it to give a lower bound on the largest size of maximal skew sets and a lower bound on the possible number of maximal skew sets.

math.AG↗
arXiv2021

Geometry of Smooth Extremal Surfaces

We study the geometry of the smooth projective surfaces that are defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible F-pure threshold among forms of the same degree. We call these surfaces $\textit{extremal surfaces}$, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially non-Frobenius split cubic surfaces of characteristic two, which are examples of extremal surfaces. For example, we show that an extremal surface $X$ contains $d^2(d^2-3d+3)$ lines where $d$ is the degree, which is notable since the number of lines on a complex surface is bounded above by a quadratic function in $d$. Whenever two of those lines meet, they determine a $d$-tangent plane to $X$ which consists of a union of $d$ lines meeting in one point; we count the precise number of such "star points" on $X$, showing that it is quintic in the degree, which recovers the fact that there are exactly 45 Eckardt points on an extremal cubic surface. Finally, we generalize the classical notion of a double six for cubic surfaces to a double $2d$ on an extremal surface of degree $d$. We show that, asymptotically in $d$, smooth extremal surfaces have at least $\frac{1}{16}d^{14}$ double $2d$'s. A key element of the proofs is using the large automorphism group of extremal surfaces which we show acts transitively on many sets, such as the set of (triples of skew) lines on the extremal surface. Extremal surfaces are closely related to finite Hermitian geometries, which we recover as the $\mathbb F_{q^2}$-rational points of special extremal surfaces defined by Hermitian forms over $\mathbb F_{q^2}$.

math.AG↗
arXiv2021

Minimal free resolutions of sheaves on the projective plane and the stable base locus decomposition of their moduli spaces

The purpose of this paper is to incorporate minimal free resolutions into the study of the birational geometry of the moduli space of coherent sheaves on the plane with character $\xi$, denoted by $M(\xi)$. We show that it is possible to recover the relevant Bridgeland destabilizing object from the minimal free resolution in order to compute the effective cone $\mathrm{Eff}(M(\xi))$ and conjecture a full relationship between Bridgeland destabilizing objects and minimal free resolutions. Moreover, we also prove that minimal free resolutions, paired with interpolation for vector bundles, yield the movable cone of the Hilbert scheme of $n$ points on the plane $\mathbb{P}^{2[n]}$, for certain values of $n$. We propose a program that computes the full stable base locus decomposition of $\mathbb{P}^{2[n]}$ based on free resolutions and interpolation. We show that this programs yields correct answers for small values of $n$.

math.AG↗
arXiv2021

Irreducibility and singularities of some nested Hilbert schemes

Let $S$ be a smooth projective surface over $\mathbb{C}$. We study the local and global geometry of the nested Hilbert scheme of points $S^{[n,n+1,n+2]}$. In particular, we show that $S^{[n,n+1,n+2]}$ is an irreducible local complete intersection with klt singularities. In addition, we compute the Picard group of $S^{[n,n+1,n+2]}$ when $h^1(S,\mathcal{O}_S) = 0$. From the irreducibility of $S^{[n,n+1,n+2]}$, we deduce irreducibility for four other infinite families of nested Hilbert schemes. We give the first explicit example of a reducible nested Hilbert scheme, which allows us to show that $S^{[n_1,\dots,n_k]}$ is reducible for $k > 22$.

math.AG↗
arXiv2021

The geometry of Hilbert schemes of two points on projective space

In this paper, we give three bases for the cohomology groups of the Hilbert scheme of two points on projective space. Then, we use these bases to compute all effective and nef cones of higher codimensional cycles on the Hilbert scheme. Next, we compute the class in one of these bases of the Chern classes of tautological bundles coming from line bundles. Finally, we provide an application of these results to the degrees of secant varieties of complete intersections.

math.AG↗
arXiv2020

Intrinsic and Extrinsic Approximation of Koopman Operators over Manifolds

This paper derives rates of convergence of certain approximations of the Koopman operators that are associated with discrete, deterministic, continuous semiflows on a complete metric space $(X,d_X)$. Approximations are constructed in terms of reproducing kernel bases that are centered at samples taken along the system trajectory. It is proven that when the samples are dense in a certain type of smooth manifold $M\subseteq X$, the derived rates of convergence depend on the fill distance of samples along the trajectory in that manifold. Error bounds for projection-based and data-dependent approximations of the Koopman operator are derived in the paper. A discussion of how these bounds are realized in intrinsic and extrinsic approximation methods is given. Finally, a numerical example that illustrates qualitatively the convergence guarantees derived in the paper is given.

math.DS↗
arXiv2020

On the position of nodes of plane curves

The Severi variety $V_{d,n}$ of plane curves of a given degree $d$ and exactly $n$ nodes admits a map to the Hilbert scheme $\mathbb{P}^{2[n]}$ of zero-dimensional subschemes of $\mathbb{P}^2$ of degree $n$. This map assigns to every curve $C\in V_{d,n}$ its nodes. For some $n$, we consider the image under this map of many known divisors of the Severi variety and its partial compactification. We compute the divisor classes of such images in Pic$(\mathbb{P}^{2[n]})$ and provide enumerative numbers of nodal curves. We also answer directly a question of Diaz-Harris about whether the canonical class of the Severi variety is effective.

math.AG↗
arXiv2018

On the birational geometry of Hilbert schemes of points and Severi divisors

We study the birational geometry of Hilbert schemes of points on non-minimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of $X^{[n]}$ and $Y^{[n]}$, when there is a birational morphism $f:X\rightarrow Y$ between surfaces. In this setting, $N^1(Y^{[n]})$ embeds in $N^1(X^{[n]})$, and we ask if the restriction of the stable base locus decomposition of $N^1(X^{[n]})$ yields the respective decomposition in $N^1(Y^{[n]})$ $i.e.$, if the weak Lefschetz Principle holds. Even though the stable base loci in $N^1(X^{[n]})$ fails to provide information about how the two decompositions interact, we show that the restriction of the augmented stable base loci of $X^{[n]}$ to $Y^{[n]}$ is equal to the stable base locus decomposition of $Y^{[n]}$. We also exhibit effective divisors induced by Severi varieties. We compute the classes of such divisors and observe that in the case that $X$ is the projective plane, these divisors yield walls of the SBLD for some cases.

math.AG↗
arXiv2017

Nef cones of nested Hilbert schemes of points on surfaces

Let $X$ be the projective plane, a Hirzebruch surface, or a general $K3$ surface. In this paper, we study the birational geometry of various nested Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes on $X$. We calculate the nef cone for two types of nested Hilbert schemes: $X^{[n+1,n]}$ and universal families. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces.

math.AG↗
arXiv2016

The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface

Let $\xi$ be a stable Chern character on $\mathbb{P}^1 \times \mathbb{P}^1$, and let $M(\xi)$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^1 \times \mathbb{P}^1$ with Chern character $\xi$. In this paper, we provide an approach to computing the effective cone of $M(\xi)$ after showing that it is a Mori dream space for all $\xi$. We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(\xi)$ which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on $\mathbb{P}^1 \times \mathbb{P}^1$.

math.AG↗