Action of $ \mathbb{G}_m $ on $ \mathbb{A}^n $ by multiplication.

Question

Consider the action of $ \mathbb{G}_m $ on $ \mathbb{A}^n $ by scalar multiplication : $ t.(a_1 , \dots , a_n ) = ( ta_1 , \dots , ta_n ) $.

In this case, there are two types of orbits :

• punctured lines through the origin.

• the origin.

Please, why does every orbit contain the origin in its closure ? and why is the origin a closed orbit ?

Thanks in advance for your help.