apply suggestions from code review

serverless/pages/manage-your-project-rest-api.mdx

@@ -7,7 +7,7 @@ tags: [ 'serverless', 'project', 'manage', 'rest', 'api']
 
 <DocBadge template="technical preview" />
 
-You can manage serverless projects using Elastic Cloud's [Project Management](https://docs.elastic.co/api-reference/project-management) REST API. This API allows you to create, update, and delete projects, as well as manage project features and usage.
+You can manage serverless projects using the [Elastic Cloud Serverless REST API](https://www.elastic.co/docs/api/doc/elastic-cloud-serverless). This API allows you to create, update, and delete projects, as well as manage project features and usage.
 
 <DocCallOut color="success" title="Tip">
     More APIs let you interact with data, capabilities, and settings inside of specific projects. Refer to the [Serverless API reference page](https://docs.elastic.co/api-reference).
@@ -43,7 +43,7 @@ curl -H "Authorization: ApiKey essu_..." https://api.elastic-cloud.com/api/v1/se
 
 The Project Management API is documented using the [OpenAPI Specification](https://en.wikipedia.org/wiki/OpenAPI_Specification). The current supported version of the specification is `3.0`.
 
-For details, check the [API reference](https://docs.elastic.co/api-reference) or download the [OpenAPI Specification](https://docs.elastic.co/api-reference/openapi/serverless-project-api.yml).
+For details, check the [API reference](https://www.elastic.co/docs/api/doc/elastic-cloud-serverless) or download the [OpenAPI Specification](https://www.elastic.co/docs/api/doc/elastic-cloud-serverless.yaml).
 
 This specification can be used to generate client SDKs, or on tools that support it, such as the [Swagger Editor](https://editor.swagger.io).
 

serverless/pages/sign-up.mdx

@@ -9,7 +9,7 @@ There are two options to create serverless projects:
 
 - If you are an existing customer, [log in to Elastic Cloud](https://cloud.elastic.co/login). On the home page, you will see a new option to create serverless projects. Note that if you are already subscribed to Elastic Cloud, there is no specific trial for serverless projects.
 
-- If you are a new user, you can [sign up for a free 14-day trial](https://cloud.elastic.co/registration), and you will be able to launch a serverless project.
+- If you are a new user, you can [sign up for a free 14-day trial](https://cloud.elastic.co/serverless-registration), and you will be able to launch a serverless project.
 
 ## What is included in my trial?