相关论文: Painlev\'e VI equations with algebraic solutions a…
Let P^2_r be the projective plane blown up at r generic points. Denote by E_0,E_1,...,E_r the strict transform of a generic straight line on P^2 and the exceptional divisors of the blown-up points on P^2_r respectively. We consider the…
We use the middle convolution to obtain some old and new algebraic solutions of the Painlev\'e VI equations.
In many cases involving mirror symmetry, it is important to have concrete mirror dual families. In this paper, we construct a natural crepant resolutions of the Batyrev-Borisov mirror dual family to the complete intersection of two cubic…
A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first…
We calculate the Picard group of generic (very general) spectral varieties living in the total space of a very ample line bundle over an algebraically closed field $k$ of odd characteristics or characteristic 0. We follow the strategy of…
Curved A-infinity algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A-infinity algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras…
In this paper, the Painlev\'e property to fractional differential equations (FDEs) are extended and the existence and uniqueness theorems for both linear and nonlinear FDEs are established. The results contribute to the research of…
Andr\'e used Hodge-theoretic methods to show that in a smooth proper family X to B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic…
Unexpected oscillations in $a_p$ values in a family of elliptic curves were observed experimentally by He, Lee, Oliver, and Pozdnyakov. We propose a heuristic explanation for these oscillations based on the "explicit formula" from analytic…
We show that the infinite staircases which arise in the ellipsoid embedding functions of rigid del Pezzo surfaces (with their monotone symplectic forms) can be entirely explained in terms of rational sesquicuspidal symplectic curves.…
It has been noticed since around 2007 that certain enumeration problems can be solved when an analytic or algebraic curve is identified. This curve is the key to the problem. In these lectures, a few such examples are presented. One is a…
In this article we are interested in the differential geometric properties of certain higher direct images of exterior powers of the sheaf of relative differentials twisted with a line bundle. We obtain explicit curvature formulas,…
We solve the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a {\em complete convergent normal form} for this class of ODEs. The normal form is optimal in the sense…
Binary geometries have recently been introduced in particle physics in connection with stringy integrals. In this work, we study a class of simple polytopes, called \emph{pellytopes}, whose number of vertices are given by Pell's numbers. We…
We study the center problem for the class $\mathcal E_\Gamma$ of Abel differential equations $\frac{dv}{dt}=a_1 v^2+a_2 v^3$, $a_1,a_2\in L^\infty ([0,T])$, such that images of Lipschitz paths $\tilde A:=\bigl(\int_0^\cdot a_1(s)ds,…
In 1957 Atiyah classified simple and indecomposable vector bundles on an elliptic curve. In this article we generalize his classification by describing the simple vector bundles on all reduced plane cubic curves. Our main result states that…
I study the solutions of a particular family of Painlev\'e VI equations with the parameters $\beta=\gamma=0, \delta=1/2$ and $2\alpha=(2\mu-1)^2$, for $2\mu\in\interi$. I show that the case of half-integer $\mu$ is integrable and that the…
We consider all \emph{odd} fundamental discriminants $D \equiv 2 \bmod 3$ and their mirror discriminants $D' = -3D$, and we study the family of elliptic curves $E_{D'}: y^{2} = x^{3} + 16D'$. We denote by $r_{3}(D)$ and $r_{3}(D')$ the rank…
In this paper, we construct higher-order generalizations of the $A_6^{(1)}$- and $A_4^{(1)}$-surface type $q$-Painlev\'e equations from the system of partial difference equations with the consistency around a cube property by periodic…
We find all solutions of the Painlev\'e VI equations with the property that they have no zeros, no poles, no 1-points and no fixed points.