Hello,
you should replace the to="abc_serialnumberid" with the column name of the case entity unless you gave the name as abc_serialnumberid.
As suggested, you need to change the case to incident. Here is a fetch xml for quick reference.
fetch version="1.0" output-format="xml-platform" mapping="logical" distinct="false">
<entity name="incident">
<attribute name="title" />
<attribute name="ticketnumber" />
<attribute name="createdon" />
<attribute name="incidentid" />
<attribute name="caseorigincode" />
<attribute name="new_serialnumber" />
<order attribute="title" descending="false" />
<link-entity name="new_serialnumber" from="new_serialnumberid" to="new_serialnumber" link-type="inner" alias="ad">
<filter type="and">
<condition attribute="new_name" operator="eq" value="{Serial Number(Case)}" />
</filter>
</link-entity>
</entity>
</fetch>
Best Regards
PG