In order to study the singularities of a variety X, we take a resolution Y--> X and compare the difference between corresponding items on each variety. One item usually used is the canonical divisor. The ``Usual discrepancy" is the difference between the canonical divisor on a resolution Y and the canonical divisor of X. This discrepancy is defined only on a normal Q-Gorenstein variety X. This discrepancy plays important roles in birational geometry. But it has a limited compatibility with the discussion of arc spaces. 

In the talk I propose another discrepancy, ``Mather-Jacobian discrepancy", which measures the difference between the canonical divisor on a resolution Y and  Omega_X^d plus Jacobian ideal of X. This discrepancy has better properties than the usual discrepancy in the  view point of arc spaces. I will show some basic properties which will be useful to study  singularities, even non-normal or non Q-Gorenstein singularities. 
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