Mathematics

In this article, we give the most genaral form of the quaternions algebra depending on 3-parameters. We define 3-parameter generalized quaternions (3PGQs) and study on various properties and applications. Firstly we present the definiton, the multiplication table another properties of 3PGQs such as addition-substraction, multiplication and multiplication by scalar operations, unit and inverse elements, conjugate and norm. We give matrix representation and Hamilton operators for 3PGQs. We get polar represenation, De Moivre's and Euler's formulas with the matrix representations for 3PGQs. Besides, we give relations among the powers of the matrices associated with 3PGQs. Finally, Lie group and Lie algebra are studied and their matrix representations are shown. Also the Lie multiplication and the killing bilinear form are given.

We construct an example of a smooth spatial quartic surface that contains 800 irreducible conics.

Over a perfect field, we determine the sheaf of A 1 -connected components of a class of threefolds given by the Blow-up of a variety admitting a P 1 -fibration over either an A 1 -rigid or a non-uniruled surface, along a smooth curve. As a consequence, we verify that the sheaf of A 1 -connected components for such varieties is A 1 -invariant.

We explicitly construct a component of the K-moduli space of K-polystable del Pezzo surfaces which is a smooth rational curve.

We categorify the commutation of Nakajima's Heisenberg operators P ±1 and their infinitely many counterparts in the quantum toroidal algebra U q 1 , q 2 ( g l 1 ¨ ) acting on the Grothendieck groups of Hilbert schemes. By combining our result with arXiv:1804.03645 , one obtains a geometric categorical U q 1 , q 2 ( g l 1 ¨ ) action on the derived category of Hilbert schemes. Our main technical tool is a detailed geometric study of certain nested Hilbert schemes of triples and quadruples, through the lens of the minimal model program, by showing that these nested Hilbert schemes are either canonical or semi-divisorial log terminal singularities.

In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley-Bacharach condition. In particular, by bounding the number of points satisfying the Cayley-Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two.

Given any arbitrary semi-algebraic set X , any two points in X may be joined by a piecewise C 2 path γ of shortest length. Suppose A is a semi-algebraic stratification of X such that each component of γ?�A is either a singleton or a real analytic geodesic segment in A , the question is whether γ?�A has at most finitely many such components. This paper gives a semi-algebraic stratification, in particular a cell decomposition, of a real semi-algebraic set in the plane whose open cells have this finiteness property. This provides insights for high dimensional stratifications of semi-algebraic sets in connection with geodesics.

We discuss properties of the de t S 2 map, present a few explicit computations, and give a geometrical interpretation for the condition de t S 2 (( v i,j ) 1≤i<j≤4 )=0 .

In this note we intend to look at the moduli stacks for global G -shtukas from a new perspective. We discuss a unifying interpretation of several moduli spaces (stacks) including moduli of global G -shtukas and (a variant of the) moduli of Higgs bundles. We view these spaces (stacks) as different fibers of a family over a scheme (stack) locally of finite type. We discuss (a relative version of) the local model theory for this family. We also consider the Hecke stacks over the moduli stack of G -shtukas and discuss the corresponding (motivic) Hecke operations.

We prove that a blowing-up formula for the intersection cohomology of the moduli space of rank 2 Higgs bundles over a curve with trivial determinant holds. As an application, we derive the Poincaré polynomial of the intersection cohomology of the moduli space under a technical assumption.