scholarly journals Threefold extremal curve germs with one non-Gorenstein point

2019 ◽  
Vol 83 (3) ◽  
pp. 565-612 ◽  
Author(s):  
S. Mori ◽  
Yu. G. Prokhorov
Keyword(s):  
Extremal Curve

scholarly journals Optimal Control on the Rotation Group SO (3)

2012 ◽  
Vol 28 (2) ◽  
pp. 321-328
Author(s):  
CLAUDIU C. REMSING ◽  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Elliptic Functions ◽  
Rotation Group ◽  
Jacobi Elliptic Functions ◽  
Hamilton Equations ◽  
Extremal Curve

A typical left-invariant optimal control problem on the rotation group SO (3) is investigated. The reduced Hamilton equations associated with an extremal curve are derived in a simple and elegant manner. These equations are then explicitly integrated by Jacobi elliptic functions.

scholarly journals A CLASSIFICATION OF LAGRANGIAN PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES

2015 ◽  
Vol 16 (4) ◽  
pp. 859-877 ◽  
Author(s):  
Benjamin Bakker
Keyword(s):  
Hilbert Scheme ◽  
Rational Curve ◽  
K3 Surface ◽  
Higher Dimensional ◽  
Extremal Curve ◽  
Symplectic Varieties

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^{2}=-2$. We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety $M$ deformation equivalent to a Hilbert scheme of $n$ points on a K3 surface, an extremal curve class $R\in H_{2}(M,\mathbb{Z})$ in the Mori cone is the line in a Lagrangian $n$-plane $\mathbb{P}^{n}\subset M$ if and only if certain intersection-theoretic criteria are met. In particular, any such class satisfies $(R,R)=-\frac{n+3}{2}$, and the primitive such classes are all contained in a single monodromy orbit.

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