We construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its Picard group. As a consequence, we obtain an explicit description of the Picard ...groups of the universal moduli stack of vector bundles on smooth curves and of the Schmitt's compactification over the stack of stable curves. We prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group. In particular, we give necessary and sufficient conditions for the existence of Poincaré bundles over the universal curve of an open substack of the rigidification, generalizing a result of Mestrano–Ramanan.
Let S be the collection of quadratic polynomial maps, and degree 2-rational maps whose automorphism groups are isomorphic to C2 defined over the rational field. Assuming standard conjectures of ...Poonen and Manes on the period length of a periodic point under the action of a map in S, we give a complete description of triples (f1,f2,p) such that p is a rational periodic point for both fi∈S, i=1,2. We also show that no more than three quadratic polynomial maps can possess a common periodic point over the rational field. In addition, under these hypotheses we show that two nonzero rational numbers a,b are periodic points of the map ϕt1,t2(z)=t1z+t2/z for infinitely many nonzero rational pairs (t1,t2) if and only if a2=b2.
Triangle areas in line arrangements Damásdi, Gábor; Martínez-Sandoval, Leonardo; Nagy, Dániel T. ...
Discrete mathematics,
December 2020, 2020-12-00, Volume:
343, Issue:
12
Journal Article
Peer reviewed
Open access
A widely investigated subject in combinatorial geometry, originated from Erdős, is the following. Given a point set P of cardinality n in the plane, how can we describe the distribution of the ...determined distances? This has been generalized in many directions.
In this paper we propose the following variants. What is the maximum number of triangles of unit area, maximum area or minimum area, that can be determined by an arrangement of n lines in the plane?
We prove that the order of magnitude for the maximum occurrence of unit areas lies between Ω(n2) and O(n9∕4+ε), for every ε>0. This result is strongly connected to additive combinatorial results and Szemerédi–Trotter type incidence theorems. Next we show an almost tight bound for the maximum number of minimum area triangles. Finally, we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques.
In a projective plane over a finite field, complete (k,n)-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all ...known examples are large, the construction of small ones, with k close to the order of the plane, is considered a hard problem. A natural candidate to be a small (k,n)-arc with few characters is the set Ω(C) of the points of a plane curve C of degree n (containing no linear components) such that some line meets C transversally in the plane, i.e. in n pairwise distinct points. Let C be either the Hermitian curve of degree q+1 in PG(2,q2r) with r≥1, or the rational BKS curve of degree q+1 in PG(2,qr) with q odd and r≥1. Then Ω(C) has four and seven characters, respectively. Furthermore, Ω(C) is small as both curves are either maximal or minimal. The completeness problem is investigated by an algebraic approach based on Galois theory and on the Hasse-Weil lower bound. Our main result for the Hermitian case is that Ω(C) is complete for r≥4. For the rational BKS curve, Ω(C) is complete if and only if r is even. If r is odd then the uncovered points by the (q+1)-secants to Ω(C) are exactly the points in PG(2,q) not lying in Ω(C). Adding those points to Ω(C) produces a complete (k,q+1)-arc in PG(2,qr), with k=qr+q. The above results do not hold true for r=2 and there remain open the case r=3 for the Hermitian curve, and the cases r=3,4 for the rational BKS curve. As a by product we also obtain two results of interest in the study of the Galois inverse problem for PGL(2,q).
Joint spectra of tuples of operators are subsets in complex projective space. We investigate the relationship between the geometry of the spectrum and the properties of the operators in the tuple ...when these operators are self-adjoint. In the case when the spectrum contains an algebraic hypersurface passing through an isolated spectral point of one of the operators we give necessary and sufficient geometric conditions for the operators in the tuple to have a common reducing subspace. We also address spectral continuity and obtain a norm estimate for the commutant of a pair of self-adjoint matrices in terms of the Hausdorff distance of their joint spectrum to a family of lines.
Let X be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g≥2 defined over an algebraically closed field K of odd characteristic p≥0, and let Aut(X) be the group of all ...automorphisms of X which fix K element-wise. For any a subgroup G of Aut(X) whose order is a power of an odd prime d other than p, the bound proven by Zomorrodian for Riemann surfaces is |G|≤9(g−1) where the extremal case can only be obtained for d=3 and g≥10. We prove Zomorrodian's result for any K. The essential part of our paper is devoted to extremal 3-Zomorrodian curves X. Two cases are distinguished according as the quotient curve X/Z for a central subgroup Z of Aut(X) of order 3 is either elliptic, or not. For elliptic type extremal 3-Zomorrodian curves X, we completely determine the two possibilities for the abstract structure of G using deeper results on finite 3-groups. We also show infinite families of extremal 3-Zomorrodian curves for both types, of elliptic or non-elliptic. Our paper does not adapt methods from the theory of Riemann surfaces, nevertheless it sheds a new light on the connection between Riemann surfaces and their automorphism groups.
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