Use Server app or Server Admin to allow delete access on the share point.
Many applications use a "safe save" to minimize data loss if the save process is interrupted. The process creates an updated copy of the file that will replace the old file. That way, if the process is interrupted, the previous version of the file remains intact.
There are two ways that applications may perform a safe save. One is to save the changes to a new file, rename the original, rename the new file to the original name, and finally delete the renamed (original) file. The other method is used on file systems such as HFS+ and AFP that support a file exchange mechanism. Changes are saved to a new temporary file, the file exchange mechanism is used (which "exchanges" the contents of the two files), and the temporary file now containing the contents of the original file is deleted.
Both methods involve deleting files. The file renaming in the first method is also a delete operation: The original name is deleted from the parent directory and the new name is added. This is why clients need permission to delete files in order to edit them.