string path = @"...";
string line = String.Empty;

while((line = reader.ReadLine()) != null)
  List<string> lst = File.ReadAllLines("Data.txt")
    .Where(arg => !string
    .IsNullOrWhiteSpace(arg))
    .ToList();
  lst.RemoveAll(x => x.Split('.')[0].Equals(name));
  File.WriteAllLines(path, lst);
}

string path = @"...";
string line = String.Empty;

while((line = reader.ReadLine()) != null)
  List<string> lst = File.ReadAllLines("Data.txt")
    .Where(arg => !string
    .IsNullOrWhiteSpace(arg))
    .ToList();
  lst.RemoveAll(x => x.Split('.')[0].Equals(name));
  File.WriteAllLines(path, lst);
}