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Higher order Jacobians, Hessians and Milnor algebras. (English) Zbl 1448.13003
Let \(f \in R = \mathbb C[x_0,\dots,x_n]\) be a reduced form. Associated to \(f\) are a number of objects studied in this paper. The Jacobian ideal is \(J(f) = (\frac{\partial f}{\partial x_0},\dots,\frac{\partial f}{\partial x_n})\). The associated hypersurface is \(V(f) \subset \mathbb P^n\). The generators define the polar map \(\varphi_X : \mathbb P^n \dashrightarrow \mathbb P^n\), and \(Z_X := \varphi(\mathbb P^n)\) is the polar image of \(X\). The restriction \(\varphi_X | X = \mathcal G_X\) is the Gauss map of \(X\). The Milnor algebra of \(f\) is \(R/J(F)\). Via Macaulay-Matlis duality the annihilator of \(f\) is a homogeneous Gorenstein artinian algebra \(A(f)\) in the dual ring \(Q = \mathbb C[X_0,\dots,X_n]\). The Jacobian matrix associated to the polar map is the Hessian matrix of \(f\). In this paper the authors generalize these objects to higher order, such as the \(k\)-th order Milnor algebra \(M^k(f) := R/J^k\), and describe some geometry associated to them. They then study \(A(f)\) and \(M^k(f)\), for different choices of \(f\) and \(k\). In particular, they study the Hilbert function and Lefschetz properties of \(A(f)\) and \(M^k(f)\). They paint a nice picture of how these things all fit together.
MSC:
13A02 Graded rings
05E40 Combinatorial aspects of commutative algebra
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
Software:
CoCoA
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[1] Abe, T.; Dimca, A., Splitting types of bundles of logarithmic vector fields along plane curves, Int. J. Math., 29, 1850055 (2018) · Zbl 1394.14020
[2] Arnold, VI; Gusein-Zade, SM; Varchenko, AN, Singularities of Differentiable Maps, vol 1. Monographs in Mathematics, 82 (1985), Boston: Birkhauser, Boston
[3] Bernstein, D.; Iarrobino, A., A non-unimodal graded Gorenstein Artin algebra in codimension five, Commun. Algebra, 20, 2323-2336 (1992) · Zbl 0761.13001
[4] Ciliberto, C.; Russo, F.; Simis, A., Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian, Adv. Math., 218, 1759-1805 (2008) · Zbl 1144.14009
[5] Abbott, J., Bigatti, A. M., Robbiano, L.: CoCoA: a system for doing Computations in Commutative Algebra. http://cocoa.dima.unige.it
[6] Di Biagio, L.; Postinghel, E., Apolarity, Hessian and Macaulay polynomials, Commun. Algebra, 41, 1, 226-237 (2013) · Zbl 1262.14069
[7] Dimca, A., On the dual and Hessian mappings of projective hypersurfaces, Math. Proc. Camb. Philos. Soc., 101, 461-468 (1987) · Zbl 0637.14001
[8] Dimca, A.; Popescu, D., Hilbert series and Lefschetz properties of dimension one almost complete intersections, Commun. Algebra, 44, 4467-4482 (2016) · Zbl 1439.13041
[9] Dimca, A.; Saito, M., A generalization of Griffiths’ theorem on rational integrals, Duke Math. J., 135, 303-326 (2006) · Zbl 1117.14012
[10] Dimca, A.; Sernesi, E., Syzygies and logarithmic vector fields along plane curves, J. Ecol. Polytech. Math., 1, 247-267 (2014) · Zbl 1327.14049
[11] Dimca, A.; Sticlaru, G., Hessian ideals of a homogeneous polynomial and generalized Tjurina algebras, Doc. Math., 20, 689-705 (2015) · Zbl 1342.14093
[12] Dimca, A., Sticlaru, G.: On the jumping lines of logarithmic vector fields along plane curves, arXiv: 1804.06349 · Zbl 1420.14064
[13] Garbagnati, A.; Repetto, F., A geometrical approach to Gordan-Noether’s and Franchetta’s contributions to a question posed by Hesse, Collect. Math., 60, 27-41 (2009) · Zbl 1180.14045
[14] Gondim, R., On higher Hessians and the Lefschetz properties, J. Algebra, 489, 241-263 (2017) · Zbl 1387.13037
[15] Gondim, R.; Russo, F., Cubic hypersurfaces with vanishing Hessian, J. Pure Appl. Algebra, 219, 779-806 (2015) · Zbl 1318.14039
[16] Gondim, R.; Zappalà, G., Lefschetz properties for Artinian Gorenstein algebras presented by quadrics, Proc. Am. Math. Soc., 146, 3, 993-1003 (2018) · Zbl 1409.13002
[17] Gondim, R.; Zappalà, G., On mixed Hessians and the Lefschetz properties, J. Pure Appl. Algebra, 223, 4268-4282 (2019) · Zbl 1423.13009
[18] Gordan, P.; Noether, M., Ueber die algebraischen Formen, deren Hesse’sche Determinante identisch verschwindet, Math. Ann., 10, 547-568 (1876) · JFM 08.0064.05
[19] Greuel, G-M; Pham, TH, Mather-Yau theorem in positive characteristic, J. Algebraic Geom., 26, 347-355 (2017) · Zbl 1357.14010
[20] Griffiths, P.: On the periods of certain rational integrals, I, II. Ann. Math. 90, 460-495 and 496-541 (1969) · Zbl 0215.08103
[21] Griffiths, P.; Harris, J., Algebraic geometry and local differential geometry, Ann. Sci. Ecol. Norm. Sup., 12, 355-452 (1969) · Zbl 0426.14019
[22] Harima, T.; Migliore, J.; Nagel, U.; Watanabe, J., The weak and strong Lefschetz properties for artinian K-algebras, J. Algebra, 262, 99-126 (2003) · Zbl 1018.13001
[23] Hesse, O., Zur Theorie der ganzen homogenen Functionen, J. Reine Angew. Math., 56, 263-269 (1859)
[24] Hulek, K., Stable rank 2 vector bundles on \({\mathbb{P}}^2\) with \(c_1\) odd, Math. Ann., 242, 241-266 (1979) · Zbl 0407.32013
[25] Ikeda, H., Results on Dilworth and Rees numbers of Artinian local rings, Jpn. J. Math., 22, 147-158 (1996) · Zbl 0857.13014
[26] Ilardi, G., Jacobian ideals, Arrangements and Lefschetz properties, J. Algebra, 508, 418-430 (2018) · Zbl 1390.13056
[27] Maeno, T.; Watanabe, J., Lefschetz elements of artinian Gorenstein algebras and Hessians of homogeneous polynomials, Ill. J. Math., 53, 593-603 (2009) · Zbl 1200.13031
[28] Marchesi, S., Vallès, J.: Nearly free curves and arrangements: a vector bundle point of view, arXiv:1712.04867
[29] Mather, JN; Yau, SS-T, Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math., 69, 243-251 (1982) · Zbl 0499.32008
[30] Migliore, J.; Mirò-Roig, RM; Nagel, U., Monomial ideals, almost complete intersections and the weak Lefschetz property, Trans. Am. Math. Soc., 363, 229-257 (2011) · Zbl 1210.13019
[31] Migliore, J.; Nagel, U., Survey article: a tour of the weak and strong Lefschetz properties, J. Commut. Algebra, 5, 3, 329-358 (2013) · Zbl 1285.13002
[32] Russo, F.: On the Geometry of Some Special Projective Varieties. Lecture Notes of the Unione Matematica Italiana, vol. 18. Springer (2016) · Zbl 1337.14001
[33] Sernesi, E., The local cohomology of the jacobian ring, Doc. Math., 19, 541-565 (2014) · Zbl 1314.14089
[34] Stanley, R., Hilbert functions of graded algebras, Adv. Math., 28, 57-83 (1978) · Zbl 0384.13012
[35] Zak, FL, Structure of Gauss maps, Funct. Anal. Appl., 21, 32-41 (1987) · Zbl 0623.14026
[36] Wang, Z., On homogeneous polynomials determined by their Jacobian ideal, Manuscr. Math., 146, 559-574 (2015) · Zbl 1329.14089
[37] Wang, Z.: On homogeneous polynomials determined by their higher Jacobians, arXiv: 1809.02422
[38] Watanabe, J.: The Dilworth number of Artinian rings and finite posets with rank function. In: Commutative algebra and combinatorics. Mathematical Society of Japan, pp. 303-312 (1987)
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