A note on nodal determinantal hypersurfaces

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A note on nodal determinantal hypersurfaces Sz-Sheng Wang1 Received: 12 March 2019 / Accepted: 10 January 2020 © Springer Nature B.V. 2020

Abstract We prove that a general determinantal hypersurface of dimension 3 is nodal. Moreover, in terms of Chern classes associated with bundle morphisms, we derive a formula for the intersection homology Euler characteristic of a general determinantal hypersurface. Keywords Nodal determinantal hypersurfaces · Intersection homology · Calabi–Yau · Fano Mathematics Subject Classification (2010) 14C17 · 32S60 · 14E05 · 14J32 · 14J45

1 Introduction A large and important class of varieties is the class of determinantal varieties. These are a particular case of degeneracy loci Di (σ ) associated with a morphism σ : E → F of vector bundles on M, which is locally defined by the ideal generated by all (i + 1)-minors of a matrix for σ . On the other hand, it is well known that a hypersurface x0 gδ (x0 , . . . , x4 ) + x1 f δ (x0 , . . . , x4 ) = 0 in P4 is a nodal determinantal hypersurface, that is, all its singular points are ordinary double points (ODPs). Here, gδ and f δ are general polynomials of degree δ. In this note, we generalize this to a general determinantal hypersurface of dimension 3. To state our result, we need an appropriate notion of the generality of morphisms. Suppose that M is a smooth proper variety and E , F have the same rank n+1. A morphism σ : E → F is said to be n-general if for all 0  i  n, the subset Di (σ )\Di−1 (σ ) is smooth of codimension (n + 1 − i)2 in the smooth proper variety M (see Definition 3.1). Theorem 1.1 (Theorem 4.4) Suppose that the smooth proper variety M has dimension 4. If a morphism σ is n-general, then the determinantal hypersurface Dn (σ ) is nodal, provided that it is irreducible. This gives us a way to construct 3-dimensional Calabi–Yau (or Fano) hypersurfaces with ODPs (see Examples 4.7 and 4.8) admitting a natural small resolution (see (9) and

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Sz-Sheng Wang [email protected] Shing-Tung Yau Center and School of Mathematics, Southeast University, Nanjing 211189, China

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Geometriae Dedicata

Proposition 3.6). Note that the nodal determinantal hypersurface Dn (σ ) is not Q-factorial (cf. Remark 2.10). To prove Theorem 1.1, we will give a formula for the intersection homology Euler characteristic of Dn (σ ), for a n-general morphism σ . The global topology of the determinantal hypersurface imposes strong restrictions on the singularities Dn (σ ) can have. We set L := det E ∨ ⊗ det F . Note that the maximal degeneracy locus Dn (σ ) is defined by det(σ ) associated to a global section of L . It is known that  c1 (L )(1 + c1 (L ))−1 c(TM ) χ(M|L ) := M

is the topological Euler characteristic of the smooth hypersurface defined by a global section of L (cf. [10, Example 4.2.6]). Let d := dim M  4. We let χ I H (−) denote the intersection homology Euler characteristic. As an analogy of the generalized Milnor number of singular hypersurfaces in [20], we write μ I H (Dn (σ ), M) := (−1)d (χ I H (Dn (σ )) − χ(M|L )). In terms