#ExperimentalMath

Alexander Kasprzyk

@amkasprzyk.bsky.social

1 month ago

Computations of Stable Multiplicities in the Cohomology of Configuration Space We describe an algorithm to compute the stable multiplicity of a family of irreducible representations in the cohomology of ordered configuration space of the plane. Using this algorithm, we compute the stable multiplicities of all families of irreducibles given by Young diagrams with 23 boxes or less up to cohomological degree 50. In particular, this determines the stable cohomology in cohomological degrees 0 <= 𝑖 <= 11. We prove related qualitative results and formulate some conjectures.

"Computations of Stable Multiplicities in the Cohomology of Configuration Space" by Emil Geisler. #ExperimentalMath #ConfigurationSpaces

Alexander Kasprzyk

@amkasprzyk.bsky.social

2 months ago

Positive Polytopes with Few Facets in the Grassmannian In this article we study adjoint hypersurfaces of geometric objects obtained by intersecting simple polytopes with few facets in P^5 with the Grassmannian Gr(2,4). These generalize the positive Grassmannian, which is the intersection of Gr(2,4) with the simplex. We show that if the resulting object has five facets, it is a positive geometry and the adjoint hypersurface is unique. For the case of six facets we show that the adjoint hypersurface is not necessarily unique and give an upper bound on the dimension of the family of adjoints. We illustrate our results with a range of examples. In particular, we show that even if the adjoint is not unique, a positive hexahedron can still be a positive geometry.

"Positive Polytopes with Few Facets in the Grassmannian" by Dmitrii Pavlova and Kristian Ranestad. #ExperimentalMath #AlgebraicGeometry #Polytopes

Alexander Kasprzyk

@amkasprzyk.bsky.social

2 months ago

The Unknotting Number, Hard Unknot Diagrams, and Reinforcement Learning We have developed a reinforcement learning agent that often finds a minimal sequence of unknotting crossing changes for a knot diagram with up to 200 crossings, hence giving an upper bound on the unknotting number. We have used this to determine the unknotting number of 57k knots. We took diagrams of connected sums of such knots with oppositely signed signatures, where the summands were overlaid. The agent has found examples where several of the crossing changes in an unknotting collection of crossings result in hyperbolic knots. Based on this, we have shown that, given knots K and K' that satisfy some mild assumptions, there is a diagram of their connected sum and u(K)+u(K') unknotting crossings such that changing any one of them results in a prime knot. As a by-product, we have obtained a dataset of 2.6 million distinct hard unknot diagrams; most of them under 35 crossings. Assuming the additivity of the unknotting number, we have determined the unknotting number of 43 at most 12-crossing knots for which the unknotting number is unknown.

"The Unknotting Number, Hard Unknot Diagrams, and Reinforcement Learning" by Taylor Applebaum, Sam Blackwell, Alex Davies, Thomas Edlich, András Juhász, Marc Lackenby, Nenad Tomašev, and Daniel Zheng. #ExperimentalMath #KnotTheory #AI #DeepMind #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

3 months ago

Mukai Lifting of Self-Dual Points in P^6 A set of 2n points in P^{n-1} is self-dual if it is invariant under the Gale transform. Motivated by Mukai’s work on canonical curves, Petrakiev showed that a general self-dual set of 14 points in P^6 arises as the intersection of the Grassmannian Gr(2,6) in its Plücker embedding in P^14 with a linear space of dimension 6. In this paper, we focus on the inverse problem of recovering such a linear space associated to a general self-dual set of points. We use numerical homotopy continuation to approach the problem and implement an algorithm in Julia to solve it. Along the way, we also implement the forward problem of slicing Grassmannians and use it to experimentally study the real solutions to this problem.

"Mukai Lifting of Self-Dual Points in P^6" by Barbara Betti and Leonie Kayser. #ExperimentalMath #AlgebraicGeometry #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

3 months ago

On the Irreducible Character Degrees of Symmetric Groups and their Multiplicities We consider problems concerning the largest degrees of irreducible characters of symmetric groups, and the multiplicities of character degrees of symmetric groups. Using evidence from computer experiments, we posit several new conjectures or extensions of previous conjectures, and prove a number of results. One of these is that, if n >= 21, then there are at least eight irreducible characters of S_n, all of which have the same degree, and which have irreducible restriction to A_n. We explore similar questions about unipotent degrees of GL_n(q). We also make some remarks about how the experiments here shed light on posited algorithms for finding the largest irreducible character degree of S_n.

"On the Irreducible Character Degrees of Symmetric Groups and their Multiplicities" by David A. Craven. #ExperimentalMath #RepresentationTheory #SymmetricGroups #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

3 months ago

A Random Matrix Model for a Family of Cusp Forms The Katz-Sarnak philosophy states that statistics of zeros of L-function families near the central point as the conductors tend to infinity agree with those of eigenvalues of random matrix ensembles as the matrix size tends to infinity. While numerous results support this conjecture, S. J. Miller observed that for finite conductors, very different behavior can occur for zeros near the central point in elliptic curve L-function families. This led to the creation of the excised model of Dueñez, Huynh, Keating, Miller, and Snaith, whose predictions for quadratic twists of a given elliptic curve are well fit by the data. The key ingredients are relating the discretization of central values of the L-functions to excised matrices based on the value of the characteristic polynomials at 1 and using lower order terms in the one-level density and pair-correlation statistics to adjust the matrix size. We extended this model to a family of twists of an L-function associated to a given holomorphic cuspidal newform of odd prime level and arbitrary weight. We derive the corresponding “effective” matrix size for a given form by computing the one-level density and pair-correlation statistics for a chosen family of twists, and we show there is no repulsion for forms with weight greater than 2 and principal nebentype. We experimentally verify the accuracy of the model, and as expected, our model recovers the elliptic curve model. Further, we uncover anomalous deviation from the expected symmetry for twists of forms with unitary symmetry.

"A Random Matrix Model for a Family of Cusp Forms" by Owen Barrett et al. #ExperimentalMath #NumberTheory #MathSky
www.tandfonline.com/doi/full/10....

Alexander Kasprzyk

@amkasprzyk.bsky.social

3 months ago

ACC for F-Signature: A Likely Counterexample Let k=\bar{F}_2 and let 0\neq \alpha\in k. We present a conjecture supported by computer experimentation involving the Brenner–Monsky quartic g_\alpha=\alpha x^2 y^2 + z^4 + xyz^2 + (x^3 + y^3)z \in k[[x,y,z]]. If true, this conjecture provides a formula for the Hilbert–Kunz multiplicity and F-signature of the family of four-dimensional hypersurfaces defined by uv + g_\alpha \in k[[x,y,z,u,v]] which depends on [F_2(\alpha):F_2], giving an infinite increasing chain of strict inequalities of F-signatures. Additionally, we obtain for any t\in N a formula for the Hilbert–Kunz multiplicity and F-signature of the t-parameter family of (3t+1)–dimensional hypersurfaces defined by uv + \sum_{i]1+^t g_{\alpha_i}g_{\alpha_i}(x_i,y_i,z_i).

"ACC for F-Signature: A Likely Counterexample" by Clay Adams, Theodore J. Sandstrom, and Austyn Simpson. #ExperimentalMath #Counterexample #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

3 months ago

On the Asymptotic Behavior of the Quantiles in the Gamma Distribution The asymptotic behavior of the quantiles in the gamma distribution are investigated as the shape parameter tends to zero. Some remarks about the behavior at infinity are given.

"On the Asymptotic Behavior of the Quantiles in the Gamma Distribution" by Henrik Laurberg Pedersen. #ExperimentalMath #GammaDistribution #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

4 months ago

A Billiard in an Open Circle and the Riemann Zeta Function We consider a dynamical billiard in a circle with one or two holes in the boundary, or q symmetrically placed holes. It is shown that the long-time survival probability, either for a circle billiard with discrete or with continuous time, can be written as a sum over never-escaping periodic orbits. Moreover, it is demonstrated that in both cases the Mellin transform of the survival probability with respect to the hole size has poles at locations determined by zeros of the Riemann zeta function and, in some cases, Dirichlet L functions.

"A Billiard in an Open Circle and the Riemann Zeta Function" by Leonid A. Bunimovich and Carl P. Dettmann. #ExperimentalMath #RiemannZetaFunction #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

4 months ago

Subdivisions of Hypersimplices: With a View Toward Finite Metric Spaces The secondary fan \Sigma(k,n) is a polyhedral fan which stratifies the regular subdivisions of the hypersimplices \Delta(k,n) We find new infinite families of rays of \Sigma(k,n), and we compute the fans \Sigma(2,7) and \Sigma(3,6). In the special case k=2 the fan \Sigma(2,n) is closely related to the metric fan MF(n), which forms a natural parameter space for the metric spaces on n points. So our results yield a classification of the finite metric spaces on seven points.

"Subdivisions of Hypersimplices: With a View Toward Finite Metric Spaces" by Laura Casabella, Michael Joswig, and Lars Kastner. #ExperimentalMath #Combinatorics #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

4 months ago

Data-Scientific Study of Kronecker Coefficients We take a data-scientific approach to study whether Kronecker coefficients are zero or not. Motivated by principal component analysis and kernel methods, we define loadings of partitions and use them to describe a sufficient condition for Kronecker coefficients to be nonzero. The results provide new methods and perspectives for the study of these coefficients.

We recently published Kyu-Hwan Lee's paper "Data-Scientific Study of Kronecker Coefficients" in #ExperimentalMath . #NumberTheory #DataScience
www.tandfonline.com/doi/full/10....

Alexander Kasprzyk

@amkasprzyk.bsky.social

4 months ago

Minimal Nonsolvable Bieberbach Groups Several authors have shown that there exist nonsolvable Bieberbach groups of dimension 15. In this paper, we show that this is, in fact, a minimal dimension for such groups.

"Minimal Nonsolvable Bieberbach Groups" by R. Lutowski and A. Szczepański. #ExperimentalMath #GroupTheory #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

4 months ago

Critical Lengths of Steklov Eigenvalues of Hypersurfaces of Revolution in Euclidean Space We study the Steklov problem on hypersurfaces of revolution with two boundary components in Euclidean space and focus on the phenomenon of critical length, at which a Steklov eigenvalue is maximized. In this article, we conjecture that, in any dimension, there is a finite number of infinite critical length. To investigate this, we develop an algorithm to efficiently perform numerical experiments, providing support to our conjecture. Furthermore, we prove the conjecture in dimension n = 3 and n = 4.

"Critical Lengths of Steklov Eigenvalues of Hypersurfaces of Revolution in Euclidean Space" by Antoine Métras and Léonard Tschanz. #ExperimentalMath #SpectralGeometry #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

4 months ago

Growth of Mahler Measure and Algebraic Entropy of Dynamics with the Laurent Property We consider the growth rate of the Mahler measure in discrete dynamical systems with the Laurent property, and in cluster algebras, and compare this with other measures of growth. In particular, we formulate the conjecture that the growth rate of the logarithmic Mahler measure coincides with the algebraic entropy, which is defined in terms of degree growth. Evidence for this conjecture is provided by exact and numerical calculations of the Mahler measure for a family of Laurent polynomials generated by rank 2 cluster algebras, for a recurrence of third order related to the Markoff numbers, and for the Somos-4 recurrence. Also, for the sequence of Laurent polynomials associated with the Kronecker quiver (the cluster algebra of affine type \tilde{A}_1 we prove a precise formula for the leading order asymptotics of the logarithmic Mahler measure, which grows linearly.

"Growth of Mahler Measure and Algebraic Entropy of Dynamics with the Laurent Property" by Andrew N. W. Hone. #ExperimentalMath #ClusterAlgebra #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

5 months ago

Clasper Presentations of Habegger-Lin’s Action on String Links Habegger and Lin gave a classification of the link-homotopy classes of links as the link-homotopy classes of string links modulo the actions of conjugations and partial conjugations for string links. In this paper, we calculate the actions of the partial conjugations and the conjugations explicitly for 4- and 5-component string links which gave classifications (presentations) of the link-homotopy classes of 4- and 5-component links. As an application, we can run Habegger and Lin’s algorithm which determines whether given two links are link-homotopic or not for 4- and 5-component links.

"Clasper Presentations of Habegger-Lin’s Action on String Links" by Yuka Kotorii and Atsuhiko Mizusawa. #ExperimentalMath #LinkHomotopy #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

5 months ago

Some Singular Curves in Mukai’s Model of \bar{M}_7 Mukai showed that the GIT quotient Gr(7,16)//Spin(10) is a birational model of the moduli space of Deligne-Mumford stable genus 7 curves \bar{M}_7. The key observation is that a general smooth genus 7 curve can be realized as the intersection of the orthogonal Grassmannian OG(5,10) in P^15 with a six-dimensional projective linear subspace. What objects appear on the boundary of Mukai’s model? As a first step in this study, computer calculations in Macaulay2, Magma, and Sage are used to find and analyze linear spaces yielding three examples of singular curves: a 7-cuspidal curve, the balanced ribbon of genus 7, and a family of genus 7 reducible nodal curves. Spin(10)-semistability is established by constructing and evaluating an invariant polynomial.

"Some Singular Curves in Mukai’s Model of \bar{M}_7" by David Swinarski. #ExperimentalMath #AlgebraicGeometry #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

5 months ago

Conjectural Criteria for the Most Singular Points of the Hilbert Schemes of Points We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the original problem to a problem in convex geometry. Proving either of the two conjectural statements will in particular resolve a long-standing conjecture by Briançon and Iarrobino back in the ’70s for the case of the powers of the maximal ideal. Furthermore, for specific classes of lengths, we conjecturally classify points satisfying the conjectural sufficient conditions. This in particular (conjecturally) provides many new explicit families of examples of maximum dimension tangent space at a point of the Hilbert schemes of points of lengths strictly between two consecutive tetrahedral numbers {3+k \choose 3}.

"Conjectural Criteria for the Most Singular Points of the Hilbert Schemes of Points" by Fatemeh Rezaee. #ExperimentalMath #AlgebraicGeometry #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

5 months ago

New Bounds and Progress Towards a Conjecture on the Summatory Function (-2)^{\Omega(n)} In this article, we study the summatory function W(x) = \sum_{n <= x}(-2)^{\Omega(n)} where \Omega(n) counts the number of prime factors of n, with multiplicity. We prove W(x) = O(x), and in particular, that |W(x)| < 2260x for all x >= 1. This provides new progress toward a conjecture of Sun, which asks if |W(x)| < x for all x >= 3078. To obtain our results, we also compute new explicit bounds on the Mertens function M(x). These bounds may be of independent interest. Moreover, we obtain similar results and make further conjectures that pertain to the more general function W_a(x) = \sum_{n <= x}(-a)^{\Omega(n)} for any real a > 0.

"New Bounds and Progress Towards a Conjecture on the Summatory Function (-2)^{\Omega(n)}" by Daniel R. Johnston, Nicol Leong, and Sebastian Tudzi. #ExperimentalMath #NumberTheory #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

5 months ago

Asymptotic Expansions Relating to the Lengths of Longest Monotone Subsequences of Involutions We study the distribution of the length of longest monotone subsequences in random (fixed-point free) involutions of n integers as n grows large, establishing asymptotic expansions in powers of n^{−1/6} in the general case and in powers of n^{−1/3} in the fixed-point free cases. Whilst the limit laws were shown by Baik and Rains to be one of the Tracy–Widom distributions F_β for β = 1 or β = 4, we find explicit analytic expressions of the first few expansions terms as linear combinations of higher order derivatives of F_β with rational polynomial coefficients. Our derivation is based on a concept of generalized analytic de-Poissonization and is subject to the validity of certain hypotheses for which we provide compelling (computational) evidence. In a preparatory step expansions of the hard-to-soft edge transition laws of LβE are studied, which are lifted into expansions of the generalized Poissonized length distributions for large intensities.

"Asymptotic Expansions Relating to the Lengths of Longest Monotone Subsequences of Involutions" by Folkmar Bornemann. #ExperimentalMath #RandomMatrices #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

5 months ago

Level Curves for Zhang’s Eta Function Study of the level curve Re(\eta(s))=0 for \eta(s)=\pi^{s/2}\Gamma(s/2)\zeta'(s) gives a new classification of the zeros of \zeta(s) and of \zeta'(s). We conjecture that for type 2 zeros, liminf(\beta'-1/2)log \gamma'=0 if and only if liminf(\gamma^+ - \gamma^-)log \gamma'=0, and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify 10^6 zeros of \zeta'(s) near T=10^10. The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices.

"Level Curves for Zhang’s Eta Function" by Jeffrey Stopple. #ExperimentalMath #NumberTheory #RiemannZetaFunction #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

5 months ago

Murmurations of Elliptic Curves We investigate the average value of the Frobenius trace at p over elliptic curves in a fixed conductor range with given rank. Plotting this average as p varies over the primes yields a striking oscillating pattern, the details of which vary with the rank. Based on this observation, we perform various data-scientific experiments with the goal of classifying elliptic curves according to their ranks.

"Murmurations of Elliptic Curves" by Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, and Alexey Pozdnyakov. #ExperimentalMath #Murmurations #NumberTheory #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

5 months ago

Orbits of Theta Characteristics The theta characteristics on a Riemann surface are permuted by the induced action of the automorphism group, with the orbit structure being important for the geometry of the curve and associated manifolds. We describe two new methods for advancing the understanding of these orbits, generalizing existing results of Kallel & Sjerve, allowing us to establish the existence of infinitely many curves possessing a unique invariant characteristic as well as determine the number of invariant characteristics for all Hurwitz curves with simple automorphism group. In addition, we compute orbit decompositions for a substantial number of curves with genus <= 9, allowing the identification of where current theoretical understanding falls short and the potential applications of machine learning techniques.

"Orbits of Theta Characteristics" by H. W. Braden and Linden Disney-Hogg. #ExperimentalMath #RiemannSurfaces #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

5 months ago

Generating Functions for Line Bundle Cohomology Dimensions on Complex Projective Varieties This paper explores the possibility of constructing multivariate generating functions for all cohomology dimensions of all holomorphic line bundles on certain complex projective varieties of Fano, Calabi-Yau and general type in various dimensions and Picard numbers. Most of the results are conjectural and rely on explicit cohomology computations. We first propose a generating function for the Euler characteristic of all holomorphic line bundles on complete intersections in products of projective spaces and toric varieties. This generating function is constructed by expanding the Hilbert-Poincaré series associated with the coordinate ring of the variety around all possible combinations of zero and infinity and then summing up the resulting contributions with alternating signs. Similar generating functions are proposed for the individual cohomology dimensions of all holomorphic line bundles on certain complete intersections, including examples of Mori and non-Mori dream spaces. Surprisingly, the examples studied indicate that a single generating function encodes both the zeroth and all higher cohomologies upon expansion around different combinations of zero and infinity, raising the question whether such generating functions determine the variety uniquely.

"Generating Functions for Line Bundle Cohomology Dimensions on Complex Projective Varieties" by Andrei Constantin. #ExperimentalMath #AlgebraicGeometry #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

5 months ago

Finite Groups with Geodetic Cayley Graphs A connected undirected graph is called geodetic if for every pair of vertices there is a unique shortest path connecting them. It has been conjectured that for finite groups, the only geodetic Cayley graphs are odd cycles and complete graphs. In this article we present a series of theoretical results which contribute to a computer search verifying this conjecture for all groups of size up to 1024. The conjecture is also verified for several infinite families of groups including dihedral and some families of nilpotent groups. Two key results which enable the computer search to reach as far as it does are: if the center of a group has even order, then the conjecture holds (this eliminates all 2-groups from our computer search); if a Cayley graph is geodetic then there are bounds relating the size of the group, generating set and center (which significantly cuts down the number of generating sets which must be searched).

"Finite Groups with Geodetic Cayley Graphs" by Murray Elder, Adam Piggott, Florian Stober, Alexander Thumm, and Armin Weiß. #ExperimentalMath #GraphTheory #GroupTheory #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

6 months ago

The Beauty of Random Polytopes Inscribed in the 2-Sphere Consider a random set of points on the unit sphere in R^d, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the case d = 3, for which there are elementary proofs and fascinating formulas for metric properties. In particular, we study the fraction of acute facets, the expected intrinsic volumes, the total edge length, and the distance to a fixed point. Finally we generalize the results to the ellipsoid with homeoid density.

"The Beauty of Random Polytopes Inscribed in the 2-Sphere" by Arseniy Akopyan, Herbert Edelsbrunner, and Anton Nikitenko. #ExperimentalMath #Combinatorics #ConvexPolytope #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

6 months ago

Combinatorics of Correlated Equilibria We study the correlated equilibrium polytope P_G of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes and prove that it is a semialgebraic set for any game. Using a stratification via oriented matroids, we propose a structured method for describing the possible combinatorial types of P_G, and show that for (2 x n)-games, the algebraic boundary of the stratification is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for generic (2 x 3)-games.

"Combinatorics of Correlated Equilibria" by Marie-Charlotte Brandenburg, Benjamin Hollering, and Irem Portakal. #ExperimentalMath #Combinatorics #ConvexPolytope #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

6 months ago

Kronecker Sequences with Many Distances The three gap theorem states that for any \alpha\in\mathbb{R} and N\in\mathbb{N}, the number of different gaps between consecutive n\alpha (mod 1) for n\in {1,...,N} is at most 3. Biringer and Schmidt (2008) instead consider the distance from each point to its nearest neighbour, generalizing to higher dimensions. ...

"Kronecker Sequences with Many Distances" by Carl Dettmann. #ExperimentalMath #KissingNumber #KroneckerSequence #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

6 months ago

The Variety of Polar Simplices We discuss the space VPS(Q, H) of ideals with Hilbert function H=(1,n,n,...) that are apolar to a full rank quadric Q. We prove that its components of saturated ideals are closely related to the locus of Gorenstein algebras and to the Slip component in border apolarity. We also point out an important error in ["The variety of polar simplices", Ranestad, K., Schreyer, F.-O. (2013)] and provide the necessary corrections.

"The Variety of Polar Simplices II" by Joachim Jelisiejew and Kristian Ranestad. #ExperimentalMath #AlgebraicGeometry #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

6 months ago

Rationality Proofs by Curve Counting We propose an approach for showing rationality of an algebraic variety X. We try to cover X by rational curves of certain type and count how many curves pass through a generic point. If the answer is 1, then we can sometimes reduce the question of rationality of X to the question of rationality of a closed subvariety of X. This approach is applied to the case of the so-called Ueno-Campana manifolds. Assuming certain conjectures on curve counting, we show that the previously open cases X_{4,6} and X_{5,6} are both rational. Our conjectures are evidenced by computer experiments. In an unexpected twist, existence of lattices D_6, E_8, and \Lambda_{10} turns out to be crucial.

"Rationality Proofs by Curve Counting" by Anton Mellit. #ExperimentalMath #AlgebraicGeometry #MathSky

Alexander Kasprzyk

@amkasprzyk.bsky.social

6 months ago

Hilbert Series of Generic Ideals in Products of Projective Spaces If k[x_1,...,x_n] / I = R = \sum_{i\geq 0}R_i, k a field, is a standard graded algebra, then the Hilbert series of R is the formal power series \sum_{i\geq 0}dim_k R_it^i. It is known already since Macaulay which power series are Hilbert series of graded algebras. A much harder question is which series are Hilbert series if we fix the number of generators of I and their degrees, say for ideals I=(f_1,...,f_r), deg f_i, i = 1,...,r. In some sense "most" ideals with fixed degrees of their generators have the same Hilbert series. There is a conjecture for the Hilbert series of those “generic” ideals, see below. In this article we make a conjecture, and prove it in some cases, in the case of generic ideals of fixed degrees in the coordinate ring of P^1 x P^1, which might be easier to prove.

"Hilbert Series of Generic Ideals in Products of Projective Spaces" by Ralf Fröberg. #ExperimentalMath #AlgebraicGeometry #MathSky