# Functors of Liftings of Projective Schemes

## TL;DR

This paper characterizes all liftings of a hyperplane section of a projective scheme with a given Hilbert polynomial using functorial methods, Gr"obner bases, and explicit constructions, providing a comprehensive parameter scheme.

## Contribution

It introduces a functorial framework for classifying liftings of projective schemes with fixed Hilbert polynomial, utilizing constructive algebraic methods.

## Key findings

- Parameter scheme for liftings constructed via gluing affine open subschemes.
- Explicit computational examples provided.
- Extension of classical results to equidimensional liftings.

## Abstract

A classical approach to investigate a closed projective scheme $W$ consists of considering a general hyperplane section of $W$, which inherits many properties of $W$. The inverse problem that consists in finding a scheme $W$ starting from a possible hyperplane section $Y$ is called a {\em lifting problem}, and every such scheme $W$ is called a {\em lifting} of $Y$. Investigations in this topic can produce methods to obtain schemes with specific properties. For example, any smooth point for $Y$ is smooth also for $W$. We characterize all the liftings of $Y$ with a given Hilbert polynomial by a parameter scheme that is obtained by gluing suitable affine open subschemes in a Hilbert scheme and is described through the functor it represents. We use constructive methods from Gr\"obner and marked bases theories. Furthermore, by classical tools we obtain an analogous result for equidimensional liftings. Examples of explicit computations are provided.