A new subadditivity formula for test ideals. (English) Zbl 1423.13051
Summary: We exhibit a new subadditivity formula for test ideals on singular varieties using an argument similar to [J.-P. Demailly et al., Mich. Math. J. 48, 137–156 (2000; Zbl 1077.14516)] and [N. Hara and K.-i. Yoshida, Trans. Am. Math. Soc. 355, No. 8, 3143–3174 (2003; Zbl 1028.13003)]. Any subadditivity formula for singular varieties must have a correction term that measures the singularities of that variety. Whereas earlier subadditivity formulas accomplished this by multiplying by the Jacobian ideal, our approach is to use the formalism of Cartier algebras [M. Blickle, J. Algebr. Geom. 22, No. 1, 49–83 (2013; Zbl 1271.13009)]. We also show that our subadditivity containment is sharper than ones shown previously in [S. Takagi, Am. J. Math. 128, No. 6, 1345–1362 (2006; Zbl 1109.14005)] and [E. Eisenstein, “Generalizations of the restriction theorem for multiplier ideals”, Preprint, arXiv:1001.2841]. The first of these results follows from a Noether normalization technique due to Hochster and Huneke. The second of these results is obtained using ideas from [S. Takagi, Math. Z. 259, No. 2, 321–341 (2008; Zbl 1143.13007)] and [Eisenstein, loc. cit.] to show that the adjoint ideal \(\mathscr{J}_X(A, Z)\) reduces mod \(p\) to Takagi’s adjoint test ideal, even when the ambient space is singular, provided that \(A\) is regular at the generic point of \(X\). One difficulty of using this new subadditivity formula in practice is the computational complexity of computing its correction term. Thus, we discuss a combinatorial construction of the relevant Cartier algebra in the toric setting.
MSC:
| 13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |
| 14B05 | Singularities in algebraic geometry |
| 14F18 | Multiplier ideals |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
References:
| [1] | Blickle, M., Test ideals via algebras of \(p^{- e}\)-linear maps, J. Algebraic Geom., 22, 1, 49-83 (2013), 2993047 · Zbl 1271.13009 |
| [2] | Blickle, M.; Schwede, K., \(p^{- 1}\)-linear maps in algebra and geometry, (Commutative Algebra (2013), Springer: Springer New York), 123-205, 3051373 · Zbl 1328.14002 |
| [3] | Blickle, M.; Schwede, K.; Takagi, S.; Zhang, W., Discreteness and rationality of \(F\)-jumping numbers on singular varieties, Math. Ann., 347, 4, 917-949 (2010), 2658149 · Zbl 1198.13007 |
| [4] | Blickle, M.; Schwede, K.; Tucker, K., \(F\)-signature of pairs and the asymptotic behavior of Frobenius splittings, Adv. Math., 231, 6, 3232-3258 (2012) · Zbl 1258.13008 |
| [5] | Brion, M.; Kumar, S., Frobenius Splitting Methods in Geometry and Representation Theory, Progress in Mathematics (2007), Birkhäuser: Birkhäuser Boston |
| [6] | Carvajal-Rojas, J.; Smolkin, D., The uniform symbolic topology property for diagonally \(F\)-regular algebras (2018) · Zbl 1444.13001 |
| [7] | Chou, J.; Hering, M.; Payne, S.; Tramel, R.; Whitney, B., Diagonal splittings of toric varieties and unimodularity, Proc. Am. Math. Soc., 146, 5, 1911-1920 (2018), 3767345 · Zbl 1420.14118 |
| [8] | Cox, D. A.; Little, J. B.; Schenck, H. K., Toric Varieties, Graduate Studies in Mathematics, vol. 124 (2011), American Mathematical Society: American Mathematical Society Providence, RI, 2810322 · Zbl 1223.14001 |
| [9] | Demailly, J.-P.; Ein, L.; Lazarsfeld, R., A subadditivity property of multiplier ideals, Mich. Math. J., 48, 137-156 (2000), dedicated to William Fulton on the occasion of his 60th birthday, 1786484 · Zbl 1077.14516 |
| [10] | Ein, L.; Lazarsfeld, R.; Smith, K. E., Uniform bounds and symbolic powers on smooth varieties, Invent. Math., 144, 2, 241-252 (2001), 1826369 · Zbl 1076.13501 |
| [11] | Eisenstein, E., Generalizations of the restriction theorem for multiplier ideals (2010), arXiv e-prints |
| [12] | D.R. Grayson, M.E. Stillman, Macaulay2, a software system for research in algebraic geometry.; D.R. Grayson, M.E. Stillman, Macaulay2, a software system for research in algebraic geometry. |
| [13] | Hara, N., A characterization of rational singularities in terms of injectivity of Frobenius maps, Am. J. Math., 120, 5, 981-996 (1998), 1646049 · Zbl 0942.13006 |
| [14] | Hara, N., Geometric interpretation of tight closure and test ideals, Trans. Am. Math. Soc., 353, 5, 1885-1906 (2001), 1813597 · Zbl 0976.13003 |
| [15] | Hara, N.; Watanabe, K.-I., \(F\)-regular and \(F\)-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom., 11, 2, 363-392 (2002), (English) · Zbl 1013.13004 |
| [16] | Hara, N.; Yoshida, K.-I., A generalization of tight closure and multiplier ideals, Trans. Am. Math. Soc., 355, 8, 3143-3174 (2003), 1974679 · Zbl 1028.13003 |
| [17] | Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics, vol. 52 (1977), Springer-Verlag: Springer-Verlag New York-Heidelberg, 0463157 · Zbl 0367.14001 |
| [18] | Hochster, M.; Huneke, C., \(F\)-regularity, test elements, and smooth base change, Trans. Am. Math. Soc., 346, 1, 1-62 (1994), 1273534 · Zbl 0844.13002 |
| [19] | Hochster, M.; Huneke, C., Tight closure in equal characteristic zero, Preprint, URL: |
| [20] | Hochster, M.; Huneke, C., Comparison of symbolic and ordinary powers of ideals, Invent. Math., 147, 2, 349-369 (2002), 1881923 · Zbl 1061.13005 |
| [21] | Kunz, E., Characterizations of regular local rings for characteristic \(p\), Am. J. Math., 91, 772-784 (1969), 0252389 · Zbl 0188.33702 |
| [22] | Lang, S., Algebra, Graduate Texts in Mathematics (2005), Springer: Springer New York |
| [23] | Lazarsfeld, R., Positivity for vector bundles, and multiplier ideals, (Positivity in Algebraic Geometry. II, Results in Mathematics and Related Areas. 3rd Series. Positivity in Algebraic Geometry. II, Results in Mathematics and Related Areas. 3rd Series, A Series of Modern Surveys in Mathematics, vol. 49 (2004), Springer-Verlag: Springer-Verlag Berlin), 2095472 · Zbl 1093.14500 |
| [24] | Miller, E.; Sturmfels, B., Combinatorial Commutative Algebra, Graduate Texts in Mathematics (2006), Springer: Springer New York |
| [25] | Payne, S., Frobenius splittings of toric varieties, Algebra Number Theory, 3, 1, 107-119 (2009), 2491910 · Zbl 1168.14036 |
| [26] | Schwede, K., \(F\)-adjunction, Algebra Number Theory, 3, 8, 907-950 (2009), 2587408 · Zbl 1209.13013 |
| [27] | Schwede, K., Centers of \(F\)-purity, Math. Z., 265, 3, 687-714 (2010), 2644316 · Zbl 1213.13014 |
| [28] | Schwede, K., Test ideals in non-\(Q\)-Gorenstein rings, Trans. Am. Math. Soc., 363, 11, 5925-5941 (2011), 2817415 · Zbl 1276.13008 |
| [29] | Schwede, K.; Tucker, K., A survey of test ideals, (Progress in Commutative Algebra 2 (2012), Walter de Gruyter: Walter de Gruyter Berlin), 39-99, 2932591 · Zbl 1254.13007 |
| [30] | Smith, K. E., The multiplier ideal is a universal test ideal, Commun. Algebra, 28, 12, 5915-5929 (2000), special issue in honor of Robin Hartshorne, 1808611 · Zbl 0979.13007 |
| [31] | Swanson, I.; Huneke, C., Integral Closure of Ideals, Rings, and Modules, vol. 13 (2006), Cambridge University Press · Zbl 1117.13001 |
| [32] | Takagi, S., Formulas for multiplier ideals on singular varieties, Am. J. Math., 128, 6, 1345-1362 (2006), 2275023 · Zbl 1109.14005 |
| [33] | Takagi, S., A characteristic \(p\) analogue of plt singularities and adjoint ideals, Math. Z., 259, 2, 321-341 (2008), 2390084 · Zbl 1143.13007 |
| [34] | Takagi, S., Adjoint ideals along closed subvarieties of higher codimension, J. Reine Angew. Math., 641, 145-162 (2010), 2643928 · Zbl 1193.14024 |
| [35] | Takagi, S., Adjoint ideals and a correspondence between log canonicity and \(F\)-purity, Algebra Number Theory, 7, 4, 917-942 (2013), 3095231 · Zbl 1305.14010 |
| [36] | Von Korff, M., F-signature of affine toric varieties (2011), arXiv e-prints |
| [37] | Zhang, Y., Is the tensor product of regular rings still regular, (version: 2010-12-21) |
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