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Register a new userRemark on the Garnier system in two variables
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Authors
Yusuke Sasano
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We remark on the Garnier system in two variables.
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In this note, we will compare the Garnier system in two variables with four-dimensional partial differential system in two variables with $W(D_6^{(1)})$-symmetry. Both systems are different in each compactification in the variables $q_1,q_2$, however, has same five holomorphy conditions in the variables $p_1,p_2$.
We present {it symmetric Hamiltonians} for the degenerate Garnier systems in two variables. For these symmetric Hamiltonians, we make the symmetry and holomorphy conditions, and we also make a generalization of these systems involving symmetry and holomorphy conditions inductively. We also show the confluence process among each system by taking the coupling confluence process of the Painleve systems.
The discrete power function on the hexagonal lattice proposed by Bobenko et al is considered, whose defining equations consist of three cross-ratio equations and a similarity constraint. We show that the defining equations are derived from the discrete symmetry of the Garnier system in two variables.
We find a one-parameter family of polynomial Hamiltonian system in two variables with $W({A}^{(1)}_1)$-symmetry. We also show that this system can be obtained by the compatibility conditions for the linear differential equations in three variables. We give a relation between it and the second member of the second Painleve hierarchy. Moreover, we give some relations between an autonomous version of its polynomial Hamiltonian system in two variables and the mKdV hierarchies.
Let $C$ be an irreducible, reduced, non-degenerate curve, of arithmetic genus $g$ and degree $d$, in the projective space $mathbf P^4$ over the complex field. Assume that $C$ satisfies the following {it flag condition of type $(s,t)$}: {$C$ does not lie on any surface of degree $<s$, and on any hypersurface of degree $<t$}. Improving previous results, in the present paper we exhibit a Castelnuovo-Halphen type bound for $g$, under the assumption $sleq t^2-t$ and $dgg t$. In the range $t^2-2t+3leq sleq t^2-t$, $dgg t$, we are able to give some information on the extremal curves. They are arithmetically Cohen-Macaulay curves, and lie on a flag like $Ssubset F$, where $S$ is a surface of degree $s$, $F$ a hypersurface of degree $t$, $S$ is unique, and its general hyperplane section is a space extremal curve, not contained in any surface of degree $<t$. In the case $dequiv 0$ (modulo $s$), they are exactly the complete intersections of a surface $S$ as above, with a hypersurface. As a consequence of previous results, we get a bound for the speciality index of a curve satisfying a flag condition.
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