57539: Fixed creation unnamed table when using public schema

Problem:
- Table on public schema being lost when tried to be created

Solution:
- Used db_schema_name argument to specify schema name of adbc_ingest

doc/source/whatsnew/v2.2.2.rst

@@ -24,6 +24,7 @@ Bug fixes
 ~~~~~~~~~
 - :meth:`DataFrame.__dataframe__` was showing bytemask instead of bitmask for ``'string[pyarrow]'`` validity buffer (:issue:`57762`)
 - :meth:`DataFrame.__dataframe__` was showing non-null validity buffer (instead of ``None``) ``'string[pyarrow]'`` without missing values (:issue:`57761`)
+- :meth:`DataFrame.to_sql` was failing to find the right table when using the schema argument (:issue:`57539`)
 
 .. ---------------------------------------------------------------------------
 .. _whatsnew_222.other:

pandas/io/sql.py

@@ -2380,7 +2380,9 @@ def to_sql(
             raise ValueError("datatypes not supported") from exc
 
         with self.con.cursor() as cur:
-            total_inserted = cur.adbc_ingest(table_name, tbl, mode=mode)
+            total_inserted = cur.adbc_ingest(
+                table_name=name, data=tbl, mode=mode, db_schema_name=schema
+            )
 
         self.con.commit()
         return total_inserted

pandas/tests/io/test_sql.py

@@ -1373,6 +1373,23 @@ def insert_on_conflict(table, conn, keys, data_iter):
         pandasSQL.drop_table("test_insert_conflict")
 
 
+@pytest.mark.parametrize("conn", postgresql_adbc_conn)
+def test_to_sql_on_public_schema(conn, request):
+    conn = request.getfixturevalue(conn)
+
+    test_data = DataFrame([[1, 2.1, "a"], [2, 3.1, "b"]], columns=list("abc"))
+    test_data.to_sql(
+        name="test_public_schema",
+        con=conn,
+        if_exists="append",
+        index=False,
+        schema="public",
+    )
+
+    df_out = sql.read_sql_table("test_public_schema", conn, schema="public")
+    assert test_data.equals(df_out)
+
+
 @pytest.mark.parametrize("conn", mysql_connectable)
 def test_insertion_method_on_conflict_update(conn, request):
     # GH 14553: Example in to_sql docstring