Some calculations on type II_1 unprojection

TL;DR

This paper explicitly calculates the linear relations of type II_1 unprojection for any parameter n ≥ 2 and provides the quadratic equation for n=3, with applications to algebraic geometry and computational code.

Contribution

It offers explicit formulas for type II_1 unprojection relations for all n ≥ 2 and details the quadratic case for n=3, advancing understanding of these algebraic structures.

Findings

Explicit linear relations for all n ≥ 2

Quadratic equation for n=3

Applications to algebraic geometry and computational implementation

Abstract

The type II_1 unprojection is, by definition, the generic complete intersection type II unprojection, in the sense of [Papadakis, Type II unprojection, J. Algebraic Geometry, 15 (2006) 399--414] Section 3.1, for the parameter value k = 1, and depends on a parameter n greater or equal than 2. Our main results are the explicit calculation of the linear relations of the type II_1 unprojection for any value of n greater or equal than 2 (Theorem 3.16) and the explicit calculation of the quadratic equation for the case n = 3 (Theorem 4.1). In addition, Section 5 contains applications to algebraic geometry while Section 6 contains the Macaulay 2 code for the type II_1 unprojection for the parameter value n = 3.