Smoothable zero dimensional schemes and special projections of algebraic varieties

TL;DR

This paper investigates the algebraic and geometric properties of projected Veronese varieties, zero-dimensional schemes, Cremona transformations, and Calabi-Yau threefolds, revealing their interconnected structures and degrees of defining equations.

Contribution

It provides new insights into the degrees of generators of ideals of projected Veronese varieties and explores their relation to zero-dimensional schemes and special algebraic varieties.

Findings

Degrees of generators depend on the projection center.

Connections between zero-dimensional schemes and Cremona transformations.

Implications for the geometry of Calabi-Yau threefolds.

Abstract

We study the degrees of generators of the ideal of a projected Veronese variety $v_2(\mathbb{P}^3)\subset \mathbb{P}^9$ to $\mathbb{P}^6$ depending on the center of projection. This is related to the geometry of zero dimensional schemes of length $8$ in $\mathbb{A}^4$, Cremona transforms of $\mathbb{P}^6$, and the geometry of Tonoli Calabi-Yau threefolds of degree $17$ in $\mathbb{P}^6$.