Higher dimensional PH curves

Higher-dimensional Analogues of Stable Curves

The Minimal Model Program offers natural higher-dimensional analogues of stable n-pointed curves and maps: stable pairs consisting of a projective variety X of dimension ≥ 2 and a divisor B, that should satisfy a few simple conditions, and stable maps f : (X, B) → Y . Although MMP remains conjectural in higher dimensions, in several important situations the moduli spaces of stable pairs, genera...

Degeneracy of entire curves into higher dimensional complex manifolds

Pursuing McQuillan’s philosophy in proving the Green-Griffiths conjecture for certain surfaces of general type, we deal with the algebraic degeneracy of entire curves tangent to holomorphic foliations by curves. Inspired by the recent work [PS14], we study the intersection of Ahlfors current T [f ] with tangent bundle TF of F , and derive some consequences. In particular, we introduce the defin...

On Genera of Smooth Curves in Higher Dimensional Varieties

We prove that for any smooth projective variety X of dimension ≥ 3, there exists an integer g0 = g0(X), such that for any integer g ≥ g0, there exists a smooth curve C in X with g(C) = g. Introduction It’s a very elementary fact that for any integer g ≥ 0, there exists a smooth curve C ⊂ P of genus g. It is interesting to ask what the analogous situation is when projective space is replaced by ...

Higher Dimensional

Control point based exact description of higher dimensional trigonometric and hyperbolic curves and multivariate surfaces

Using the normalized B-bases of vector spaces of trigonometric and hyperbolic polynomials of finite order, we specify control point configurations for the exact description of higher dimensional (rational) curves and (hybrid) multivariate surfaces determined by coordinate functions that are exclusively given either by traditional trigonometric or hyperbolic polynomials in each of their variable...