Marginal cost -- The portion of the marginal cost curve above its intersection with the average variable cost curve is the supply curve for a firm operating in a perfectly competitive market. (the portion of the MC curve below its intersection with the AVC curve is not part of the supply curve because a firm would not operate at price below the shutdown point) This is not true for firms operating in other market structures. For example, while a monopoly ``has'' an MC curve it does not have a supply curve. In a perfectly competitive market, a supply curve shows the quantity a seller's willing and able to supply at each price -- for each price, there is a unique quantity that would be supplied. The one-to-one relationship simply is absent in the case of a monopoly. With a monopoly, there could be an infinite number of prices associated with a given quantity. It all depends on the shape and position of the demand curve and its accompanying marginal revenue curve. 

Having read that, I wonder is the marginal cost curve the supply curve?
No
In the first paragraph, it says "The portion of the marginal cost curve above its intersection with the average variable cost curve is the supply curve for a firm operating in a perfectly competitive market."
By definition, we know that only part of MC could be used as S. Therefore they are not equal to each other and thus No..