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Perazzo 3-folds and the weak Lefschetz property. (English) Zbl 1516.14066

A Perazzo 3-fold is a hypersurface in \(\mathbb P^4\) of degree \(d\) whose equation in homogeneous coordinates \(x_0, x_1, x_2, u, v\) has the form \(f = p_0 x_0 + p_1 x_1 + p_2 x_2 + g\), with \(p_i\) algebraically dependent but linearly independent forms of degree \(d-1\) in \(u,v\) and \(g\) a degree \(d\) form in \(u,v\) [U. Perazzo, Batt. G. (2) 38, 337–354 (1900; JFM 31.0127.02)]. The Perazzo 3-folds have vanishing hessian, but in general are not cones, hence give counterexamples to an old conjecture of Hesse. Associated to \(f\) is the quotient ring \(A_f\) of the ring \(S = k[\partial/\partial x_0, \dots \partial/\partial u]\) of differential operators by the annihilator of \(f\). It is known that \(A_f\) is a standard Artinian Gorenstein algebra whose socle degree is \(d = \deg f\). There are hessians for \(f\) of each order \(0 \leq t \leq \lfloor d/2 \rfloor\) and \(A_f\) fails to strong Lefschetz property (SLP) if and only if at least one of these hessians vanishes [T. Maeno and J. Watanabe, Ill. J. Math. 53, No. 2, 591–603 (2009; Zbl 1200.13031)].
It is known that if \(d=3\), then \(A_f\) fails the WLP, but R. Gondim has shown that \(A_f\) has the WLP if \(d=4\) [J. Algebra 489, 241–263 (2017; Zbl 1387.13037)]. The authors show that for \(d > 4\), there are a maximal and minimal \(h\)-vectors \(h\) (determined uniquely by the Hilbert function) for the algebra \(A_f\) and that \(A_f\) satisfies the WLP if \(h\) is minimal, but \(A_f\) fails the WLP if \(h\) is maximal. These are the only possibilities if \(d=5\); for \(d>5\) they show by example that for \(h\) neither maximal nor minimal, \(A_f\) may or may not satisfy the WLP. Focusing on the algebras \(A_f\) with \(h\) minimal, they use the theory of Waring rank for forms in two variables to list all such Perazzo \(3\)-folds. The classification is given in terms of how the plane \(\pi = \langle p_0, p_1, p_2 \rangle \subset \mathbb P^{d-1} = \mathbb P (k[u,v]_{d-1})\) interacts with the rational normal curve \(C \subset \mathbb P^{d-1}\); (a) \(\pi\) is is an osculating plane to \(C\), (b) \(\pi\) contains a tangent line to \(C\) and meets \(C\) at a third point, or (c) \(\pi\) intersects \(C\) in three distinct points.

MSC:

14H70 Relationships between algebraic curves and integrable systems
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13N10 Commutative rings of differential operators and their modules
14M07 Low codimension problems in algebraic geometry

Software:

Macaulay2
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References:

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