A proof of the Proportionality Theorem

Proportionality of Indices of 1-Forms on Singular Varieties

This article is about 1-forms on complex analytic varieties and it is particularly relevant when the variety has non-isolated singularities. We first show how the radial extension technique of M.-H. Schwartz can be adapted to 1-forms, allowing us to define the Schwartz index of 1-forms with isolated singularities on singular varieties. Then we see how MacPherson's local Euler obstruction, adapted to 1-forms in general, relates to the Schwartz index, thus obtaining a proportionality theorem for these indices analogous to the one for vector fields. We also extend the definition of the GSV-index …

One more proof of the Abhyankar-Moh-Suzuki theorem

We extract the Abhyankar-Moh-Suzuki theorem from the Lin-Zaidenberg theorem.

A Shorter Proof of Marten's Theorem

This note is intended to give a new proof on Marten's theorem stating that $\dim(\wrd) \leq d-2r$ for any smooth curve with equality occurring exactly in the case when C is hyper-elliptic. The proof follows the general lines of the proof given in 'The Geometry of Algebraic Curves'. What is different is that it uses Hopf's theorem to simplify the proof and strengthen the conclusions of the theorem.

A toric proof of Bézout's theorem for weighted projective spaces

Using toric geometry we prove a Bézout type theorem for weighted projective spaces.

A Short Proof of a Grothendieck-Lefschetz Theorem for Equivariant Picard Groups

We give a short proof of a Grothendieck-Lefschetz Theorem for equivariant Picard groups of nonsingular varieties with the action of an affine algebraic group.

Christoph定理的代数几何证明

Algebro-geometric proof of Christoph's Theorem

本文通过代数方法证明了Christoph定理，该定理表明存在有限自动机可计算代数函数级数的系数。