Search

It is a classical result that there are 12 (irreducible) rational cubic curves through 8 generic points in P C 2 , but little is known about the non-generic cases. The space of 8-point configurations is partitioned into strata depending on combinatorial objects we call quatroids, a higher-order version of representable matroids. We compute all 779,777 quatroids on eight distinct points in the plane, which produces a full description of the stratification. For each stratum, we generate several invariants, including the number of rational cubics through a generic configuration. As a byproduct of our investigation, we obtain a collection of results regarding the base loci of pencils of cubics and positive certificates for non-rationality. [ABSTRACT FROM AUTHOR]

We study smooth curves on which the alternating group A 6 acts faithfully. Let V ⊂ PGL (3 , C) be the Valentiner group, which is isomorphic to A 6 . We see that there are integral V -invariant curves of degree 12 which have geometric genera 10 and 19. On the other hand, if A 6 acts faithfully on a curve C of genus 10 or 19, then we give an explicit description of the extension k (C / A 5) / k (C / A 6) for any icosahedral subgroup A 5 . Using this, we show the uniqueness of smooth projective curves of genera 10 and 19 whose automorphism groups contain A 6 . [ABSTRACT FROM AUTHOR]

We describe symmetries of the braid monodromy decomposition for a class of plane curves defined over reals including the real curves with no real points and proving new divisibility relations for Alexander invariants of such curves. [ABSTRACT FROM AUTHOR]

Self-intersections and local singular points of the curves play an important role in algebraic geometry and many other areas. In the present paper, we study the self-intersection and local singular points of the n-member chains. For this purpose, we derive and use several new results on trigonometric formulas. A unified approach for calculating self-intersection and local singular points for a wide class of curves is presented. An application to the spectral theory of integro-differential operators with difference kernels is given as well. [ABSTRACT FROM AUTHOR]

Holditch’s theorem is a classical geometrical result on the areas of a given closed curve and another one, its Holditch curve, which is constructed as the locus of a fixed point dividing a chord of constant length that moves with its endpoints over the given curve and that returns back to its original position after some full revolution. Holditch curves have already been studied from the parametric point of view, although numerical methods and approximations are often necessary for their computation. In this paper, implicit equations of Holditch curves of algebraic curves are studied. The implicit equations can be simply found from the computation of a resultant of two polynomials. With the same techniques, Holditch curves of two initial algebraic curves are also considered. Moreover, the use of implicit equations allows to find new and explicit parameterizations of non-trivial Holditch curves, such as in the case of having an ellipse as an initial curve. [ABSTRACT FROM AUTHOR]

There is a proposition due to Kollár as reported by Kollár (Proceedings of the summer research institute, Santa Cruz, CA, USA, July 9–29, 1995, American Mathematical Society, Providence, 1997) on computing log canonical thresholds of certain hypersurface germs using weighted blowups, which we extend to weighted blowups with non-negative weights. Using this, we show that the log canonical threshold of a convergent complex power series is at most 1/c, where (c , ... , c) is a point on a facet of its Newton polyhedron. Moreover, in the case n = 2 , if the power series is weakly normalised with respect to this facet or the point (c, c) belongs to two facets, then we have equality. This generalises a theorem of Varchenko 1982 to non-isolated singularities. [ABSTRACT FROM AUTHOR]

We consider the geometric map C , called Cayleyan, associating to a plane cubic E the adjoint of its dual curve. We show that C and the classical Hessian map H generate a free semigroup. We begin the investigation of the geometry and dynamics of these maps, and of the geometrically special elliptic curves: these are the elliptic curves isomorphic to cubics in the Hesse pencil which are fixed by some endomorphism belonging to the semigroup W (H , C) generated by H , C . We point out then how the dynamic behaviours of H and C differ drastically. Firstly, concerning the number of real periodic points: for H these are infinitely many, for C they are just 4. Secondly, the Julia set of H is the whole projective line, unlike what happens for all elements of W (H , C) which are not iterates of H . [ABSTRACT FROM AUTHOR]

We study factorizations of HOMFLY polynomials of certain prime knots and oriented links. We begin with a computer analysis of prime knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for HOMFLY polynomials of oriented links associated to 2-connected plane graphs. [ABSTRACT FROM AUTHOR]

We study subsets S of curves X whose double structure does not impose independent conditions to a linear series L, but there are divisors D ∈ | L | singular at all points of S. These subsets form the Terracini loci of X. We investigate Terracini loci, with a special look towards their non-emptiness, mainly in the case of canonical curves, and in the case of space curves. [ABSTRACT FROM AUTHOR]

We introduce a notion of good cohomology for multiple lines in P 3 and we classify multiple lines with good cohomology up to multiplicity 4. In particular, we show that the family of space curves of degree d, not lying on a surface of degree < d , and of maximal arithmetic genus is not irreducible already for d = 4 and d = 5 . [ABSTRACT FROM AUTHOR]

This paper presents a new characterisation of the Fermat curve, according to the arrangement of Galois points. [ABSTRACT FROM AUTHOR]

In this work, we use the finiteness of the Mordell–Weil group of the Jacobian variety of the curve C : y 2 = x (x - 3) (x - 4) (x - 6) (x - 7) and the Riemann Roch spaces to determine explicitly the set of algebraic points of given degree l over Q on the curve C . The results obtained extend the work of Gordon and Grant, who determined the Mordell–Weil group J (Q) and the set of rational points on the same curve. [ABSTRACT FROM AUTHOR]

The 16-year old Blaise Pascal found an incidence relation that holds when six points lie on a conic. A century later, Braikenridge and Maclaurin extended Pascal's result to a straightedge construction that characterizes when six points lie on a conic. Nearly 400 years later, we develop a straightedge construction to check whether ten points lie on a cubic curve. [ABSTRACT FROM AUTHOR]

Let X = X 1 ∪ ⋯ ∪ X s ⊂ P n , n ≥ 4 , be a general union of smooth non-special curves with X i of degree d i and genus g i and d i ≥ max { 2 g i - 1 , g i + n } if g i > 0 . We prove that X has maximal rank, i.e., for any t ∈ N either h 0 (I X (t)) = 0 or h 1 (I X (t)) = 0 if it is so in a few explicit cases in P 4 . We also prove an unconditional weaker result, maximal rank up to a positive integer δ n . [ABSTRACT FROM AUTHOR]

We study the multigraded Hilbert function of general rational curves of prescribed multidegree contained in a multiprojective space Y. We have strong negative results, e.g. for almost all line bundles L on Y there is a multidegree such that maximal rank with respect to L fails for all smooth rational curves C of that multidegree, i.e. h 0 (I C ⊗ L) > 0 and h 1 (I C ⊗ L) > 0 . This is different from the case of general rational curves in projective spaces. We also have positive results, most of them being for the case Y = P n × P 1 , n ≥ 2 . [ABSTRACT FROM AUTHOR]

In recent years, many useful applications of the polynomial method have emerged in finite geometry. Indeed, algebraic curves, especially those defined by Rédei-type polynomials, are powerful in studying blocking sets. In this paper, we reverse the engine and study when blocking sets can arise from rational points on plane curves over finite fields. We show that irreducible curves of low degree cannot provide blocking sets and prove more refined results for cubic and quartic curves. On the other hand, using tools from number theory, we construct smooth plane curves defined over F p of degree at most 4 p 3 / 4 + 1 whose points form blocking sets. [ABSTRACT FROM AUTHOR]

In the paper Targino (Outer Lipschitz Geometry of Complex Algebraic Plane Curves. International Mathematics Research Notices, rnac202. https://doi.org/10.1093/imrn/rnac202, 2022) Renato Targino shows that bi-Lipschitz type of a plane curve is determined by the local ambient topological properties of curves. Here we show that it is no longer true in higher dimensions. However we show that the bi-Lipschitz type of space curves is determined by the number of singular points and by the local ambient topological type of a generic projection of such curves into the affine plane. [ABSTRACT FROM AUTHOR]

We prove that on X n , the plane blown-up at n very general points, there are Ulrich line bundles with respect to a line bundle corresponding to curves of degree m passing simply through the n blown-up points, with m ⩽ 2 n and such that the line bundle in question is very ample on X n . We prove that the number of these Ulrich line bundles tends to infinity with n. We also prove the existence of slope-stable rank-r Ulrich vector bundles on X n , for n ⩾ 2 and any r ⩾ 1 and we compute the dimensions of their moduli spaces. These computations imply that X n is Ulrich wild. [ABSTRACT FROM AUTHOR]

Poncelet's theorem states that if there exists an n-sided polygon which is inscribed in a given conic C and circumscribed about another conic D, then there are infinitely many such n-gons. Proofs of this theorem that we are aware of, including Poncelet's original proof and the celebrated modern proof by Griffiths and Harris, assume the two conics to be in general position (that is, not tangent or at least not osculating), or be defined over C , or both. Here we show that Poncelet's theorem holds for any two conics C and D in the projective plane P 2 over an algebraically closed field k of any characteristic other than two. If C and D are osculating and char k > 2 , our result shows that there always exist infinitely many polygons of char k sides that are inscribed in C and circumscribed about D. We also describe the situation in characteristic two in the appendix. [ABSTRACT FROM AUTHOR]

Creating a formal model for timbre is one of the most compelling open questions in music research. In contrast to more traditional perceptually-oriented approaches, often aimed at sound analysis, we introduce a three-dimensional geometric model of timbre, specifically designed for sound synthesis. The proposed model relies on the properties of space-filling curves for multidimensional scaling, and represents via three parameters, any static combination of sinusoidal partials with an additional noisiness component. We detail the construction of the model and its properties and discuss future implications for music research. [ABSTRACT FROM AUTHOR]

We determine the minimal bi-degree(s) of an irreducible filling curve over F q for ℙ 1 × ℙ 1 by constructing examples. It is (q + 1 , q + 1) if q ≠ 2 , and they are (4, 3) and (3, 4) if q = 2. [ABSTRACT FROM AUTHOR]

Smooth algebraic plane quartics over algebraically closed fields of characteristic different than two have 28 bitangent lines. Their tropical counterparts often have infinitely many bitangents. They are grouped into seven equivalence classes, one for each linear system associated to an effective tropical theta characteristic on the tropical quartic. We show such classes determine tropically convex sets and provide a complete combinatorial classification of such objects into 41 types (up to symmetry). The occurrence of a given class is determined by both the combinatorial type and the metric structure of the input tropical plane quartic. We use this result to provide explicit sign-rules to obtain real lifts for each tropical bitangent class, and confirm that each one has either zero or exactly four real lifts, as previously conjectured by Len and the second author. Furthermore, such real lifts are always totally-real. [ABSTRACT FROM AUTHOR]

Divergence free vector fields are called magnetic vector fields in three-dimensional semi-Riemannian manifolds. When a charged particle enters the magnetic vector field, it traces a new trajectory called magnetic curve by the influenced of magnetic field. In the present paper, we investigate the magnetic curves on the lightlike surfaces corresponding to the Killing magnetic fields in 3D semi-Riemannian manifolds. Moreover, we give some characterizations of these curves. As an application, we determine all magnetic curves on the lightlike cone. Finally, we give various examples to confirm the main results. [ABSTRACT FROM AUTHOR]

For an irreducible conic C in a Desarguesian plane of odd square order, estimating the number of points, from a Baer subplane, which are external to C is a natural problem. In this paper, a complete list of possibilities is determined for the case where C shares at least one point with the subplane. [ABSTRACT FROM AUTHOR]

Let S be a smooth algebraic surface in P 3 (C) . Movasati and Sertöz (Rend. Circ. Mat. Palermo 2:1–17, 2020) associate an ideal I α (C) to the primitive cohomology class α (C) of C in S. We show that the equations of C can be determined by I α (C) under numerical conditions. We apply this result to reconstruct rational curves and arithmetically Cohen-Macaulay curves from their cohomology classes. On the other hand, we show that the class α (C) of a rational quartic curve C on a smooth quartic surface S is not even perfect, that is, that I α (C) is bigger than the sum of the Jacobian ideal of S and of the homogeneous ideals of curves D in S for which I α (D) = I α (C) . [ABSTRACT FROM AUTHOR]

In this paper, we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell S M TNN in the totally non-negative Grassmannian G r TNN (k , n) to the spectral data for the relevant class of real regular Kadomtsev–Petviashvili II (KP-II) solutions, thus completing the search of real algebraic-geometric data for the KP-II equation started in Abenda and Grinevich (Commun Math Phys 361(3):1029–1081, 2018; Sel Math New Ser 25(3):43, 2019). The spectral curve is modeled on the Krichever construction for degenerate finite-gap solutions and is a rationally degenerate M -curve, Γ , dual to the graph. The divisors are real regular KP-II divisors in the ovals of Γ , i.e. they fulfill the conditions for selecting real regular finite-gap KP-II solutions in Dubrovin and Natanzon (Izv Akad Nauk SSSR Ser Mat 52:267–286, 1988). Since the soliton data are described by points in S M TNN , we establish a bridge between real regular finite-gap KP-II solutions (Dubrovin and Natanzon, 1988) and real regular multi-line KP-II solitons which are known to be parameterized by points in G r TNN (k , n) (Chakravarty and Kodama in Stud Appl Math 123:83–151, 2009; Kodama and Williams in Invent Math 198:637–699, 2014). We use the geometric characterization of spaces of relations on plabic networks introduced in Abenda and Grinevich (Adv Math 406:108523, 2022; Int Math Res Not 2022:rnac162, 2022. ) to prove the invariance of this construction with respect to the many gauge freedoms on the network. Such systems of relations were proposed in Lam (in: Current developments in mathematics, International Press, Somerville, 2014) for the computation of scattering amplitudes for on-shell diagrams N = 4 SYM (Arkani-Hamed et al. in Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge, 2016) and govern the totally non-negative amalgamation of the little positive Grassmannians, G r TP (1 , 3) and G r TP (2 , 3) , into any given positroid cell S M TNN ⊂ G r TNN (k , n) . In our setting they control the reality and regularity properties of the KP-II divisor. Finally, we explain the transformation of both the curve and the divisor both under Postnikov’s moves and reductions and under amalgamation of positroid cells, and apply our construction to some examples. [ABSTRACT FROM AUTHOR]

Chang and Ran proved, in 1984, the unirationality of the moduli spaces of curves of genus 11, 12, and 13. They consider a family U of nonsingular space curves. If U is non-empty, it dominates the moduli space. Considering certain vector bundles related to the curves, one shows that U is unirational. Finally, Chang and Ran prove that U is non-empty by constructing smoothable reducible curves. We prove, instead, the existence of (almost) globally generated vector bundles of the above kind. The dependency locus of (rank − 1) general global sections of such a bundle is a nonsingular curve from U. [ABSTRACT FROM AUTHOR]

The discriminant of a smooth plane cubic curve over the complex numbers can be written as a product of theta functions. This provides an important connection between algebraic and analytic objects. In this paper, we perform a new approach to obtain this classical result by using determinantal representations. More precisely, one can represent a non-singular cubic form as the determinant of a matrix whose elements are linear forms. Theta functions naturally appear in this representation and thus in the discriminant of the cubic. [ABSTRACT FROM AUTHOR]

A criterion for the existence of a plane model of an algebraic curve such that the Galois closures of projections from two points are the same is presented. Surprisingly, the Hermitian function field in positive characteristic becomes the Galois closure of projections of a plane curve from two non-uniform points. [ABSTRACT FROM AUTHOR]

First we give a sharp upper bound for the cardinal m of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve C. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve C, in terms of its degree d and of the minimal degree r ≤ d - 1 of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for m is also attained. A second characterization of these curves in terms of the 0-th Fitting ideal of their Jacobian module is also given. Finally we prove the existence of curves with maximal global Tjurina numbers for certain pairs (d, r). We conjecture that such curves exist for any pair (d, r), and that, in addition, they may be chosen to be line arrangements when r ≤ d - 2 . This conjecture is proved for degrees d ≤ 11 . [ABSTRACT FROM AUTHOR]

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology. [ABSTRACT FROM AUTHOR]