Test, multiplier and invariant ideals

Transcription

1 Test, multiplier and invariant ideals Inês B. Henriques Levico Terme MOCCA 2014 Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 1 / 39

2 Motivation: Measures of Singularities Measuring of Singularities using multiplicity Measuring of Singularity Multiplicity Let f be a polynomial over a field C, vanishing at z C n. We say f is smooth at z iff f 0 for some i. x i z For simplicity assume z = 0. Define the multiplicity, or order of singularity of f at z to be max {d every differential of order d, f z = 0, } Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 2 / 39

3 Motivation: Measures of Singularities Measuring of Singularities using multiplicity Measures of Singularity Multiplicity The multiplicity is an important first step in the study of singularities, but it is a coarse measure. The following curves of multiplicity two manifest different levels of singularity. yx =0 y 2 = x 3 y 2 = x 17 Test ideals in algebra, as well as multiplier ideals in birational geometry and analysis, spurred from an effort to better measure singularities. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 3 / 39

4 Motivation: Measures of Singularities 3 approaches 2 characteristics 2 invariants Measuring Singularities 3 approaches - 2 characteristics - 2 invariants These far more subtle measurements emerged from analytic, geometric and algebraic points of view. char K = 0, the log canonical threshold, lct can be defined analytically (via integration), or geometrically (via resolution of singularities). char K = p > 0, the F-pure threshold, fpt can be defined algebraically using the Frobenius endomorphism. Remarkably, lct and fpt define essentially the same invariant. The smaller the values of these invariants, the worse the singularities. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 4 / 39

5 Motivation: Measures of Singularities 3 approaches 2 characteristics 2 invariants Measuring Singularities Analytic approach to the Log canonical threshold f C[x 1,..., x n ]. Analytically: lct (f) = smallest real number λ > 0 s. t. f 2λ is not locally integrable = sup {λ R + B ε(0) } 1 <, for some ε > 0. f 2λ Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 5 / 39

6 Motivation: Measures of Singularities 3 approaches 2 characteristics 2 invariants Measuring Singularities Analytic approach: f = z 1 a 1... z n a n then lct (f) = min i { } 1 a i Example Let f = z 1 a 1... z n a n, a i N. Use polar coordinates to integrate f z i = r i, det = r 1... r n : 1 r 1... r n z 1 2a1λ dµ = 2anλ... z n r 2a 1 1 λ... r 2a dr n nλ Fubini s theorem: 1 r 2λ a r 2λ a n n 1 B ε(0) iff 2λ a i 1 < 1 iff λ < min i a i dr < for some ε > 0 { } 1 = lct (f). Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 6 / 39

7 Motivation: Measures of Singularities 3 approaches 2 characteristics 2 invariants Measuring Singularities Analytic approach: f is smooth at 0 then lct(f) = 1. Example If f is smooth at 0 then f can be taken to be part of a system of local coordinates for C n at the origin, thus 1 dµ < iff λ < 1. 2λ B ε(0) f lct(f) = 1 Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 7 / 39

8 Motivation: Measures of Singularities 3 approaches 2 characteristics 2 invariants Measuring Singularities Algebraic approach to the F-pure threshold f Z p [x 1,..., x n ] where p is prime. For e 0, let ν f (p e ) := max{r 0 f r / (x 1 p e,..., x n p e ) }. The F-pure threshold of f is fpt (f) := lim e ν f (p e ) p e. the limit exists and is contained in (0, 1] Q. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 8 / 39

9 f = x 2 + y 3 Motivation: Measures of Singularities 3 approaches 2 characteristics 2 invariants Example Let f Z 2 [x, y] with p = 2, and f = x 2 + y 3. ν f (2) = max{r 0 (x 2 + y 3 ) r (x, y) [2] = (x 2, y 2 )} = 0 ν f (2 3 ) = max{r 0 (x 2 + y 3 ) r (x, y) [8] = (x 8, y 8 )} = 3 = ν f (2 e ) = max{r 0 (x 2 + y 3 ) r (x 2e, y 2e )} = 2 e 1 1 fpt (f p ) = lim e 2 e e = 1/2 Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 9 / 39

10 Motivation: Measures of Singularities 2 characteristics: Log canonical threshold and F-threshold Asymptotically F-threshold = log canonical threshold Fix f Z[x 1,..., x n ]. compute lct (f) Or reduce modulo p f p Z p [x 1,..., x n ] fpt (f p ) Theorem (Hara-Yoshida 2003) (1) For all primes p, fpt (f p ) lct (f); (2) lim p fpt (f p) = lct (f). Conjecture (Mustata-Takagi-Watanabe 2005) Does equality holds for infinitely many primes? Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 10 / 39

11 f = x 2 + y 3 Motivation: Measures of Singularities 2 characteristics: Log canonical threshold and F-threshold Example (Mustata-Takagi-Watanabe 2005) Let f = x 2 + y 3. f C[x 1,..., x n ] lct (f) = 5 6 f p F p [x 1,..., x n ] 1/2 if p = 2 2/3 if p = 3 fpt (f p ) = 5/6 if p 1 mod 6 5/6 1 if p 5 mod 6 6p lim fpt (f) = 5/6 = lct (f). p There are infinitely many p for which fpt (f p ) = lct (f). Work of Elkies shows that fpt (f p ) lct (f) for infinitely many primes. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 11 / 39

12 Motivation: Measures of Singularities 2 characteristics: Log canonical threshold and F-threshold f = elliptic curve Example ([Bhatt, 2013]) Let f define an elliptic curve E in P 2 over Z. f C[x 1,..., x n ] lct (f) = 1 f p F p [x 1,..., x n ] E P 2 F p fpt (f p ) = { 1 if E is ordinary, 1 1 p if E is supersingular. There are infinitely many p for which fpt (f p ) = lct (f). Work of Elkies shows that fpt (f p ) lct (f) for infinitely many primes. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 12 / 39

13 Generalized Test ideals, F-jumping numbers and F-thresholds The Frobenius endomorphism The Frobenius endomorphism and e th -root Let S = K[x 1,..., x s ] with char K = p (K = Z p ) and I = (f 1,..., f r ) be an ideal. Fix e 0, q = p e. Definition (1) F e : S S, defined by F e (g) = g pe = g q, for g S, denotes the e th Frobenius endomorphism. (2) The images of S and I are denoted S q = {g q g S} and I [q] := F e (I) = (f q 1,..., f q r ). (3) One easily sees that B e = { x u xus s 0 u 1,..., u s q 1} forms a basis for S over S q Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 13 / 39

14 Generalized Test ideals, F-jumping numbers and F-thresholds The Frobenius endomorphism The e th -root Let S = K[x 1,..., x s ] with char K = p, I an ideal, and fix e 0 q = p e. Definition The e th -root ideal of I, denoted I [1/q] to be the smallest ideal J such that I J [q]. Remark If we express the generators of I = (f 1,..., f s ) B e = { x u xus s 0 u 1,..., u s q 1} by in terms of f i = µ B e g q iµ µ, for some polynomial g iµ, for i = 1,..., s, then, the q th -root of the coefficients generate I [1/q] : I [1/q] = ( g iµ µ B, i = 1,..., s ). Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 14 / 39

15 Generalized Test ideals, F-jumping numbers and F-thresholds Generalized test ideals Generalized test ideals Let I be an ideal and e 0, q as before. Definition Given λ R +, the generalized test ideal of I at λ is τ (λ I ) = (I ) [1/p λpe e ] = (I ) [1/p λpe e ] for e 0. e 0 Properties It defines a non-increasing, right continuous family of ideals associated to I, in the sense that: λ λ, τ (λ I ) τ ( λ I ) ; λ ε > 0 : τ (λ I ) = τ ( λ I ), λ [λ, λ + ε). Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 15 / 39

16 Generalized Test ideals, F-jumping numbers and F-thresholds Jumping numbers and F-threshold Jumping numbers and F-threshold Definition The points of discontinuity, or jumps, λ R + for which τ (λ I ) τ (λ δ I ) for all δ > 0, are called F-jumping numbers of the ideal I. fpt (I) := min{λ R + τ (λ I ) S} first jump. Theorem (Blickle-Mustata-Smith 2008) The set of jumping numbers is a discrete subset of Q. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 16 / 39

17 Generalized Test ideals, F-jumping numbers and F-thresholds Multiplier ideals Multiplier ideal: Analytically Definition Given f C[x 1,..., x N ] vanishing at z C N, the log-canonical threshold of f at z is defined as: 1 lct z (f) = sup{λ R + : <, a ball z B} f 2λ Example: lct 0 (x a 1 1 xa N N ) = min i {1/a i }. B Definition Given an ideal I = (f 1,..., f r ) S and λ R +, the multiplier ideal with coefficient λ of I is defined as { } g J (λ I) := g S : ( r i=1 f i 2 ) λ L1 loc, where L 1 loc denotes the space of locally integrable functions. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 17 / 39

18 Generalized Test ideals, F-jumping numbers and F-thresholds Multiplier ideals Multiplier ideal: Geometrically Definition Alternatively, given an ideal I S = K[x 1,..., x N ], char(k) = 0 and λ R +, the multiplier ideal with coefficient λ of I is J (λ I) = π O X (K X/ Spec(S) λ F ) where: (i) π : X Spec(S) is a log-resolution of the sheafication Ĩ of I; (ii) π 1( Ĩ ) = O X ( F ). (iii) K X/ Spec(S) is the relative canonical divisor. This simply means that X is non-singular, F is an effective divisor, the exceptional locus E of π is a divisor and F + E has simple normal crossing support. Log-resolutions like this, in characteristic 0, always exist by Hironaka s celebrated result. The log-canonical threshold of an ideal I S is: lct (I) = min{λ R + : J (λ I) S}. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 18 / 39

19 Generalized Test ideals, F-jumping numbers and F-thresholds Multiplier ideals Multiplier ideals Fix I = (g 1,..., g n ) S = K[x 1,..., x N ], char(k) = 0. If g i Z, I R is said to be reduced from characteristic zero to, in characteristic p > 0. I p = I Z Z p Z p [x 1,..., x n ] In characteristic zero, the multiplier ideals of I associate to I, a right continuous, non-decreasing family of ideals {J (λ I)} λ 0, where the first jumping number defines the Log canonical threshold of the ideal I, lct (I). Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 19 / 39

20 Generalized Test ideals, F-jumping numbers and F-thresholds Asymptotically Generalized test ideals = Multiplier ideals Asymptotically: Generalized test ideals = Multiplier ideals Theorem (Hara-Yoshida 2003) Assume that I S is an ideal reduced from characteristic zero to characteristic p > 0. (1) For all λ and all primes p, τ (λ I ) J (λ I); (2) Fixing λ, one has τ (λ I ) = J (λ I) for p 0. Conjecture (Mustata-Takagi-Watanabe 2005) Does equality hold (for all λ) for infinitely many primes? Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 20 / 39

21 Determinantal ideals lct and fpt of a determinantal ideal Determinantal ideals Given 1 t m n, consider the m n matrix of indeterminates over a field K x 11 x 12 x 1n x 21 x 22 x 2n X = x m1 x m2 x mn Let I = I t (X m n ) S = K[x 11,..., x mn ], the ideal generated by t-minors of the X. Theorem (Johnson 2004, Miller-Singh-Varbaro 2013) [Johnson] If char K = 0, { (n k)(m k) } lct (I) = min k = 0,..., t 1. (t k) [MSV] If char K = p > 0, fpt (I) = lct (I) for all prime p. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 21 / 39

22 Determinantal ideals Multiplier ideals of a determinantal ideal Multiplier ideals of maximal determinantal ideals Let I t = I t (X m n ) S = K[x 11,..., x mn ] for all 1 t m. Theorem [Johnson, 2004] Let char K = 0. The multiplier ideal of I m at λ, is for all λ lct (I). λ lct (I)+1 J (λ I m ) = Im Theorem [Johnson, 2004] Let char K = 0 and 1 t m. The multiplier ideal of I t for all λ lct (I). J (λ I t ) = m i=1 I ( λ (t i+1) +1 (m i+1)(n i+1)) i at λ satisfies Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 22 / 39

23 Determinantal ideals Generalized test ideals of a determinantal ideal Test ideals of arbitrary determinantal ideals Theorem[H.] Let R = k[x 1,..., x mn ] char k = p > 0 and I = I m (X m n ). For all λ fpt (I), τ (λ I ) = I λ fpt (I)+1 = J (λ I) Z Z p (in all prime characteristics p). The set of F -jumping numbers of I is fpt (I) + N. Conjecture[H.-Varbaro] For all 1 t m n let I = I t (X m n ), τ (λ I ) = J (λ I) Z Z p, for all λ. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 23 / 39

24 Determinantal ideals Generalized test ideals of a determinantal ideal Notation Given 1 t m n, consider the m n matrix X of indeterminates over a field K. Let K[X] be the polinomial ring in the entries of X and I t K[X] be the prime ideal generated by the t-minors of X. Fix two K-vector spaces, V and W, with dim V = m and dim W = n, and consider the group G = GL(V ) GL(W ). Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 24 / 39

25 Determinantal ideals Generalized test ideals of a determinantal ideal The action of G on K[X] Fixing basis for V and W, one can consider the action of G on K[X] defined by (A, B) X = AXB 1 (A, B) G = GL(V ) GL(W ). With respect to this action, I t is an invariant ideal. In characteristic 0, De Concini, Eisenbud e Procesi [DEP] gave a description of the invariant ideals of K[X]. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 25 / 39

26 Determinantal ideals Generalized test ideals of a determinantal ideal Young diagrams A (Young) diagram is a vector σ = (σ 1,..., σ k ) with positive integers as entries, such that σ 1 σ 2... σ k 1. We write σ = (r s 1 1, rs 2 2,...) to denote the tuple with first s 1 entries equal to r 1, the following s 2 entries of σ are equal to r 2 and so on... Given diagrams σ = (σ 1,..., σ k ) and τ = (τ 1,..., τ h ), we write σ τ if k h and σ i τ i for all i = 1,..., k. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 26 / 39

27 Determinantal ideals Generalized test ideals of a determinantal ideal Cauchy formula Consider the natural diagonal action of G on V W, the following isomorphism is G-equivariant: K[X] = Sym(V W ) = d 0 Sym d (V W ) If char(k) = 0, the Cauchy formula yields a decomposition of Sym(V W ) into irreducible G-modules Sym(V W ) = σ S σ V S σ W, where σ is a Young diagram with σ 1 m functor. and S σ denotes the Schur E.g., when σ = (d), S (d) V = S (1 d )V = Sym d V. d V and when σ = (1 d ) = (1,..., 1) Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 27 / 39

28 Determinantal ideals Generalized test ideals of a determinantal ideal The G-invariant ideals of Sym(V W ). Therefore, the G-invariant vector spaces of Sym(V W ) correspond to such sets Σ of Young diagrams: Σ σ Σ S σ V S σ W. It is shown in [DEP] that such a vector space is an ideal iff: σ Σ, τ σ τ Σ. Further, the ideal I σ generated by S σ V S σ W admits the decomposition: I σ = S τ V S τ W τ σ Note: The determinantal ideal I t corresponds to the ideal I (t). Given a (finite) set Σ of Young diagrams, set: I(Σ) = σ Σ I σ. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 28 / 39

29 Determinantal ideals Generalized test ideals of a determinantal ideal γ-functions Thus, in charateristic 0, the G-invariant ideals of Sym(V W ) correspond to finite sets Σ of Young diagrams: Σ I(Σ). For each i {1,..., m}, define the γ i function on Young diagrams by: γ i (σ) = k max{0, σ j i + 1}, for σ = (σ 1,..., σ k ). j=1 Given a finite set of diagrams, Σ, we define the polytope P Σ R m as the convex hull of {(γ 1 (σ),..., γ m (σ)) : σ Σ}. We give an explicit description of the multiplier ideals of I(Σ) in terms of the polytope P Σ. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 29 / 39

30 Determinantal ideals Generalized test ideals of a determinantal ideal The multiplier ideals of I(Σ) when Σ = {σ}; In this talk we focus on the description of the multiplier ideals of I σ In other words, we describe the multiplier ideals, and consequently the log canonical thresholds, of I σ for each Young diagram σ. In the case where σ = (t), I σ = I t is a determinantal ideal, we recover a result of [Johnson, 2003] using log-resolutions of determinantal varieties. Recently, [DoCampo, 2012] recovered the formula for the log-canonical threshold of I t from the study of jet schemes associated to determinantal varieties. Given a product of minors = δ 1 δ k K[X], where δ i is a α i -minor of X, we define α = (α 1,..., α k ) to be the shape of. We will see that the multiplier ideals of I σ are generated by products of minors of prescribed shapes (described by γ-functions). Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 30 / 39

31 Determinantal ideals Generalized test ideals of a determinantal ideal Theorem [H.-Varbaro] Given a Young diagram σ = (σ 1,..., σ k ) with σ 1 m, and λ R +, J (λ I σ ) is generated by products of minors whose shape α satisfies: γ i (α) λγ i (σ) + 1 (m i + 1)(n i + 1) i = 1,..., m. Equivalently, J (λ I σ ) = m i=1 I ( λγ i(σ) +1 (m i+1)(n i+1)) i In particular, the log-canonical threshold of I σ is given by { } (m i + 1)(n i + 1) lct (I σ ) = min. i γ i (σ) Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 31 / 39

32 Determinantal ideals Generalized test ideals of a determinantal ideal Proof We work towards developing a theory that gives a description of the test ideals (hence the F -pure thresholds) of ideals with certain nice properties, in a polynomial over a field K of positive characteristic. We recover a description of the multiplier ideals, from a result of [Hara-Yoshida, 2003]: the test ideals at λ of the characteristic p reduction = p > 0 characteristic p reduction of the multiplier ideals at λ Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 32 / 39

33 Determinantal ideals Generalized test ideals of a determinantal ideal G-invariant ideals in positive characteristic In positive characteristic, there isn t a characterization of the G-invariant ideals of K[X]. A priori, there isn t an obvious way to define the ideals I σ, in positive characteristic. But we still know how to handle enough G-invariant ideals, even in positive characteristic, to do the job! Given a Young diagram σ = (σ 1,..., σ k ) with σ 1 m, we set D σ = I σ1 I σk K[X]. Theorem[DEP] If char(k) = 0, then I σ = D σ. Corollary: For each λ R +, J (λ I σ ) = J (λ D σ ). Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 33 / 39

34 Determinantal ideals τ(λ I) = e>0 Generalized test ideals of a determinantal ideal Test ideals Recall/Definitions: Let S = K[x 1,..., x N ] be a polynomial ring over a field K, char(k) = p > 0. Given an ideal I = (f 1,..., f r ) and q = p e, recall that: (1) I [q] = (f q 1,..., f q r ) S; (2) I [1/q] denotes the smallest (!) ideal J S for which I J [q] ; (3) the test ideal of I at λ( R + ) is (I λq ) [1/q] Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 34 / 39

35 Determinantal ideals Generalized test ideals of a determinantal ideal Test ideals Theorem[Hara-Yoshida] Given I P = Z[x 1,..., x N ] and λ R +, there exists a prime p 0 such that: J (λ I P Z C) P Z Z/pZ = τ(λ I P Z Z/pZ) We computed the test ideals (hence F -pure thresholds) of every ideal of the form D σ (products of determinantal ideals, cf. [H.-Varbaro]) It is worth noting that these are the same in all characteristics. Recall that the F -pure threshold of determinantal ideals (σ = (t)) was known, cf. [MSV]. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 35 / 39

36 Determinantal ideals Generalized test ideals of a determinantal ideal Big test ideals Definition Given an ideal I S = K[x 1,..., x N ] and p Spec(S), define the function f I;p : Z >0 Z >0 by: f I;p (s) = max{l : I s p (l) }. f I;p is linear, f I;p (s) = s f I;p (1), so we set: e p (I) = f I;p (1) = max{l : I p (l) }. Proposition If K has positive characteristic, then: τ(λ I) p ( λep(i) +1 ht(p)) λ R +. ( ) p Spec(S) Definition An ideal I S has big test ideals if equality holds in ( ), for all λ R +. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 36 / 39

37 Determinantal ideals Generalized test ideals of a determinantal ideal Condition ( ) Naturally, for each s Z >0, one has: I s p Spec(S) p (ep(i)s). To make this inclusion optimal we introduce the following condition: Definition: An ideal I S satisfies condition ( ) if for all p Spec(S) there exists a function g I;p : Z >0 Z >0 for which: I s = p (g I;p(s)) s 0. (1) p Spec(S) Remark: One easily sees that, for some c N, one has e p (I)s c g I;p (s) e p (I)s. Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 37 / 39

38 Determinantal ideals Generalized test ideals of a determinantal ideal Condition ( +) A result of [Bruns] shows that, for every Young diagram σ, the ideal D σ satisfies condition ( ). Indeed: D s σ = m i=1 I (γ i(σ)s) i Definition: An ideal I S satisfies condition ( +) if it satisfies condition ( ), and there exists a term order on S and a polynomial F S such that: (i) in (F ) is a square-free monomial; (ii) F p (ht(p)) for all p ) (I s. s Z >0 Ass Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 38 / 39

39 Determinantal ideals Generalized test ideals of a determinantal ideal Proposition For all Young diagram σ, the ideal D σ satisfies ( +). Theorem[H.-Varbaro] If I S satisfies condition ( +), then it has big test ideals, i.e. τ(λ I) = p ( λep(i) +1 ht(p)) λ R +. ( ) p Spec(S) Corollary τ(λ D σ ) is generated by products of minors whose shape α satisfies: γ i (α) λγ i (σ) + 1 (m i + 1)(n i + 1) i = 1,..., m. Equivalently, τ(λ D σ ) = m i=1 I ( λγ i(σ) +1 (m i+1)(n i+1)) i. In particular, the F -pure threshold is given by: { } (m i + 1)(n i + 1) fpt (D σ ) = min. i γ i (σ) Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 39 / 39

40 Bibliography B. Bhatt, The F-pure threshold of an elliptic curve, bhattb/math/cyfthreshold.pdf A. Benito, E. Faber, K. E. Smith, Measuring Singularities with Frobenius: The Basics, Commutative algebra. Expository papers dedicated to David Eisenbud on the occasion of his 65th birthday. Edited by Irena Peeva. Springer, New York, (2013). viii+707 pp. M. Blickle, M. Mustata, K. Smith, Discreteness and rationality of F-thresholds. Special volume in honor of Melvin Hochster. Michigan Math. J. 57 (2008), pp W. Bruns, Algebras defined by powers of determinantal ideals, J. Algebra 142 (1991), C. DeConcini, D. Eisenbud, C. Procesi, Young diagrams and determinantal varieties, Invent. Math. 56 (1980), R. Docampo, Arcs on determinantal varieties, Trans. Amer. Math. Soc. 365 (2012) Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 39 / 39

41 Bibliography N. Hara, K. I. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc., 355 (2003), no. 8, I. B. Henriques, F-thresholds and test ideals of determinantal ideals of maximal minors, preprint arxiv: I. B. Henriques and M. Varbaro, Test, multiplier and invariant ideals, preprint arxiv: A. A. Johnson, Multiplier ideals of determinantal ideals, Ph.D. Thesis, University of Michigan (2003). L. Miller, A. Singh, M. Varbaro The F-threshold of a determinantal ideal. preprint arxiv: M. Mustata, S. Takagi, K. Watanabe F-thresholds and Bermstein Sato polynomials. European Congress of Mathematics,pp , Eur. Math. Soc., Zurich (2005). Inês B. Henriques (Levico Terme) Test, multiplier and invariant ideals 39 / 39