Motivation Discreteness and rationality on rings with singularities What about the non-(log)- Q -Gorenstein case? Discreteness and rationality of F -jumping numbers on rings with singularities Karl Schwede 1 , Wenliang Zhang 1 , Shunsuke Takagi 2 1 Department of Mathematics University of Michigan 2 Department of Mathematics Kyushu University Sectional AMS Meeting – Spring 2009 Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Discreteness and rationality on rings with singularities What about the non-(log)- Q -Gorenstein case? Outline Motivation 1 Multiplier ideals Test ideals Discreteness and rationality on rings with singularities 2 What about the non-(log)- Q -Gorenstein case? 3 Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Multiplier ideals Discreteness and rationality on rings with singularities Test ideals What about the non-(log)- Q -Gorenstein case? Outline Motivation 1 Multiplier ideals Test ideals Discreteness and rationality on rings with singularities 2 What about the non-(log)- Q -Gorenstein case? 3 Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Multiplier ideals Discreteness and rationality on rings with singularities Test ideals What about the non-(log)- Q -Gorenstein case? Outline Motivation 1 Multiplier ideals Test ideals Discreteness and rationality on rings with singularities 2 What about the non-(log)- Q -Gorenstein case? 3 Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Multiplier ideals Discreteness and rationality on rings with singularities Test ideals What about the non-(log)- Q -Gorenstein case? Multiplier ideals on singular varieties Suppose that X = Spec R is normal (of finite type / C ). We let ∆ be an effective Q -divisor. A Q -divisor is a linear combination of subvarieties of codimension 1 such with positive rational coefficients. We also assume that K X + ∆ is Q -Cartier. Here K X is a divisor in the whose divisor class corresponds to ω X . Q -Cartier means that there exists some n ∈ Z such that n ∆ is integral (all denominators were cleared) and nK X + n ∆ is Cartier (ie, locally trivial in the divisor class group). When X is Q -Gorenstein (that means nK X is Cartier, locally ω ( n ) ∼ = R , for some n ), we can choose ∆ = 0. X Then for any ideal a on X , the setting of a triple ( X , ∆ , a t ) (for t ∈ R ≥ 0 ) is the natural context for considering multiplier ideals from the point of view of the “MMP”. Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Multiplier ideals Discreteness and rationality on rings with singularities Test ideals What about the non-(log)- Q -Gorenstein case? Multiplier ideals on singular varieties Suppose that X = Spec R is normal (of finite type / C ). We let ∆ be an effective Q -divisor. A Q -divisor is a linear combination of subvarieties of codimension 1 such with positive rational coefficients. We also assume that K X + ∆ is Q -Cartier. Here K X is a divisor in the whose divisor class corresponds to ω X . Q -Cartier means that there exists some n ∈ Z such that n ∆ is integral (all denominators were cleared) and nK X + n ∆ is Cartier (ie, locally trivial in the divisor class group). When X is Q -Gorenstein (that means nK X is Cartier, locally ω ( n ) ∼ = R , for some n ), we can choose ∆ = 0. X Then for any ideal a on X , the setting of a triple ( X , ∆ , a t ) (for t ∈ R ≥ 0 ) is the natural context for considering multiplier ideals from the point of view of the “MMP”. Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Multiplier ideals Discreteness and rationality on rings with singularities Test ideals What about the non-(log)- Q -Gorenstein case? Multiplier ideals on singular varieties Suppose that X = Spec R is normal (of finite type / C ). We let ∆ be an effective Q -divisor. A Q -divisor is a linear combination of subvarieties of codimension 1 such with positive rational coefficients. We also assume that K X + ∆ is Q -Cartier. Here K X is a divisor in the whose divisor class corresponds to ω X . Q -Cartier means that there exists some n ∈ Z such that n ∆ is integral (all denominators were cleared) and nK X + n ∆ is Cartier (ie, locally trivial in the divisor class group). When X is Q -Gorenstein (that means nK X is Cartier, locally ω ( n ) ∼ = R , for some n ), we can choose ∆ = 0. X Then for any ideal a on X , the setting of a triple ( X , ∆ , a t ) (for t ∈ R ≥ 0 ) is the natural context for considering multiplier ideals from the point of view of the “MMP”. Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Multiplier ideals Discreteness and rationality on rings with singularities Test ideals What about the non-(log)- Q -Gorenstein case? Multiplier ideals on singular varieties Suppose that X = Spec R is normal (of finite type / C ). We let ∆ be an effective Q -divisor. A Q -divisor is a linear combination of subvarieties of codimension 1 such with positive rational coefficients. We also assume that K X + ∆ is Q -Cartier. Here K X is a divisor in the whose divisor class corresponds to ω X . Q -Cartier means that there exists some n ∈ Z such that n ∆ is integral (all denominators were cleared) and nK X + n ∆ is Cartier (ie, locally trivial in the divisor class group). When X is Q -Gorenstein (that means nK X is Cartier, locally ω ( n ) ∼ = R , for some n ), we can choose ∆ = 0. X Then for any ideal a on X , the setting of a triple ( X , ∆ , a t ) (for t ∈ R ≥ 0 ) is the natural context for considering multiplier ideals from the point of view of the “MMP”. Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Multiplier ideals Discreteness and rationality on rings with singularities Test ideals What about the non-(log)- Q -Gorenstein case? Multiplier ideals on singular varieties Suppose that X = Spec R is normal (of finite type / C ). We let ∆ be an effective Q -divisor. A Q -divisor is a linear combination of subvarieties of codimension 1 such with positive rational coefficients. We also assume that K X + ∆ is Q -Cartier. Here K X is a divisor in the whose divisor class corresponds to ω X . Q -Cartier means that there exists some n ∈ Z such that n ∆ is integral (all denominators were cleared) and nK X + n ∆ is Cartier (ie, locally trivial in the divisor class group). When X is Q -Gorenstein (that means nK X is Cartier, locally ω ( n ) ∼ = R , for some n ), we can choose ∆ = 0. X Then for any ideal a on X , the setting of a triple ( X , ∆ , a t ) (for t ∈ R ≥ 0 ) is the natural context for considering multiplier ideals from the point of view of the “MMP”. Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Multiplier ideals Discreteness and rationality on rings with singularities Test ideals What about the non-(log)- Q -Gorenstein case? Multiplier ideals on singular varieties Suppose that X = Spec R is normal (of finite type / C ). We let ∆ be an effective Q -divisor. A Q -divisor is a linear combination of subvarieties of codimension 1 such with positive rational coefficients. We also assume that K X + ∆ is Q -Cartier. Here K X is a divisor in the whose divisor class corresponds to ω X . Q -Cartier means that there exists some n ∈ Z such that n ∆ is integral (all denominators were cleared) and nK X + n ∆ is Cartier (ie, locally trivial in the divisor class group). When X is Q -Gorenstein (that means nK X is Cartier, locally ω ( n ) ∼ = R , for some n ), we can choose ∆ = 0. X Then for any ideal a on X , the setting of a triple ( X , ∆ , a t ) (for t ∈ R ≥ 0 ) is the natural context for considering multiplier ideals from the point of view of the “MMP”. Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Multiplier ideals Discreteness and rationality on rings with singularities Test ideals What about the non-(log)- Q -Gorenstein case? Multiplier ideals on singular varieties Suppose that X = Spec R is normal (of finite type / C ). We let ∆ be an effective Q -divisor. A Q -divisor is a linear combination of subvarieties of codimension 1 such with positive rational coefficients. We also assume that K X + ∆ is Q -Cartier. Here K X is a divisor in the whose divisor class corresponds to ω X . Q -Cartier means that there exists some n ∈ Z such that n ∆ is integral (all denominators were cleared) and nK X + n ∆ is Cartier (ie, locally trivial in the divisor class group). When X is Q -Gorenstein (that means nK X is Cartier, locally ω ( n ) ∼ = R , for some n ), we can choose ∆ = 0. X Then for any ideal a on X , the setting of a triple ( X , ∆ , a t ) (for t ∈ R ≥ 0 ) is the natural context for considering multiplier ideals from the point of view of the “MMP”. Karl Schwede, Wenliang Zhang, Shunsuke Takagi
Motivation Multiplier ideals Discreteness and rationality on rings with singularities Test ideals What about the non-(log)- Q -Gorenstein case? The definition of multiplier ideals Take a log resolution π : � X → X with a O � X = O � X ( − E ) . I’m not going to give a precise definition here. Then (using this Q -Cartier notion), we can define the multiplier ideal J ( X , ∆ , a t ) to be X − π ∗ ( K X + ∆) − tE ⌉ ) . π ∗ O � X ( ⌈ K � The round-up just rounds up the coefficients of the Q -divisors. Another way to think of this is that there are a finite number of discrete valuations v i (of Frac R ) and integers m i and n i > 0 such that J ( X , ∆ , a t ) = { r ∈ R | v i ( r ) ≥ ⌊ n i t + m i ⌋} Here n i is just the order of a along v i and the m i depend on ∆ , v i and the singularities of X . Karl Schwede, Wenliang Zhang, Shunsuke Takagi
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