Bernstein--Sato polynomials of locally quasi-homogeneous divisors in $\mathbb{C}^{3}$
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| Format: | Preprint |
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2024
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| _version_ | 1866913590670262272 |
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| author | Bath, Daniel |
| author_facet | Bath, Daniel |
| contents | We consider the Bernstein--Sato polynomial of a locally quasi-homogeneous polynomial $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$. We construct, in the analytic category, a complex of $\mathscr{D}_{X}[s]$-modules that can be used to compute the $\mathscr{D}_{X}[s]$-dual of $\mathscr{D}_{X}[s] f^{s-1}$ as the middle term of a short exact sequence where the outer terms are well understood. This extends a result by Narváez Macarro where a freeness assumption was required.
We derive many results about the zeroes of the Bernstein--Sato polynomial. First, we prove each nonvanishing degree of the zeroeth local cohomology of the Milnor algebra $H_{\mathfrak{m}}^{0} (R / (\partial f))$ contributes a root to the Bernstein--Sato polynomial, generalizing a result of M. Saito's (where the argument cannot weaken homogeneity to quasi-homogeneity). Second, we prove the zeroes of the Bernstein--Sato polynomial admit a partial symmetry about $-1$, extending a result of Narváez Macarro that again required freeness. We give applications to very small roots, the twisted Logarithmic Comparison Theorem, and more precise statements when $f$ is additionally assumed to be homogeneous.
Finally, when $f$ defines a hyperplane arrangement in $\mathbb{C}^{3}$ we give a complete formula for the zeroes of the Bernstein--Sato polynomial of $f$. We show all zeroes except the candidate root $-2 + (2 / \text{deg}(f))$ are (easily) combinatorially given; we give many equivalent characterizations of when the only non-combinatorial candidate root $-2 + (2/ \text{deg}(f))$ is in fact a zero of the Bernstein--Sato polynomial. One equivalent condition is the nonvanishing of $H_{\mathfrak{m}}^{0}( R / (\partial f))_{\text{deg}(f) - 1}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_08342 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bernstein--Sato polynomials of locally quasi-homogeneous divisors in $\mathbb{C}^{3}$ Bath, Daniel Algebraic Geometry Commutative Algebra Combinatorics Complex Variables 32S20, 32S22 (Primary) 32C38, 14F10, 32S40, 14B15, 13D45 (Secondary) We consider the Bernstein--Sato polynomial of a locally quasi-homogeneous polynomial $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$. We construct, in the analytic category, a complex of $\mathscr{D}_{X}[s]$-modules that can be used to compute the $\mathscr{D}_{X}[s]$-dual of $\mathscr{D}_{X}[s] f^{s-1}$ as the middle term of a short exact sequence where the outer terms are well understood. This extends a result by Narváez Macarro where a freeness assumption was required. We derive many results about the zeroes of the Bernstein--Sato polynomial. First, we prove each nonvanishing degree of the zeroeth local cohomology of the Milnor algebra $H_{\mathfrak{m}}^{0} (R / (\partial f))$ contributes a root to the Bernstein--Sato polynomial, generalizing a result of M. Saito's (where the argument cannot weaken homogeneity to quasi-homogeneity). Second, we prove the zeroes of the Bernstein--Sato polynomial admit a partial symmetry about $-1$, extending a result of Narváez Macarro that again required freeness. We give applications to very small roots, the twisted Logarithmic Comparison Theorem, and more precise statements when $f$ is additionally assumed to be homogeneous. Finally, when $f$ defines a hyperplane arrangement in $\mathbb{C}^{3}$ we give a complete formula for the zeroes of the Bernstein--Sato polynomial of $f$. We show all zeroes except the candidate root $-2 + (2 / \text{deg}(f))$ are (easily) combinatorially given; we give many equivalent characterizations of when the only non-combinatorial candidate root $-2 + (2/ \text{deg}(f))$ is in fact a zero of the Bernstein--Sato polynomial. One equivalent condition is the nonvanishing of $H_{\mathfrak{m}}^{0}( R / (\partial f))_{\text{deg}(f) - 1}$. |
| title | Bernstein--Sato polynomials of locally quasi-homogeneous divisors in $\mathbb{C}^{3}$ |
| topic | Algebraic Geometry Commutative Algebra Combinatorics Complex Variables 32S20, 32S22 (Primary) 32C38, 14F10, 32S40, 14B15, 13D45 (Secondary) |
| url | https://arxiv.org/abs/2402.08342 |