[Paper Review] Blowups and Resolution

H. Hauser|arXiv (Cornell University)|Apr 3, 2014

Algebraic Geometry and Number Theory69 references4 citations

TL;DR

This paper provides a concise, dictionary-style overview of blowups and resolution of singularities in algebraic geometry, focusing on foundational concepts, techniques, and invariants. It presents key results on ideal transforms, maximal contact, and resolution in zero and positive characteristic, with explicit examples illustrating the failure of maximal contact in positive characteristic and the role of differential operators.

ABSTRACT

This article shall serve as a quick reference for somebody who needs precise information on concepts and results related to resolution of singularities. As such, it is more a technical manual than a bedtime story. Topics which are covered: Singular and regular points of varieties and schemes; various definitions of blowups and their mutual relations; properties of blowups; transforms of varieties, schemes and ideals; exceptional divisors; Cartier and normal crossings divisors; transversality; hypersurfaces of maximal contact; flags; coefficient ideals; resolution invariants; order of ideals; Hilbert-Samuel function; semicontinuity; various resolution statements; characteristic zero resolution; characteristic p phenomena. The text is complemented with lots of illustrating examples.

Motivation & Objective

• To provide a rapid, accessible reference for researchers in algebraic geometry and commutative algebra on blowups and resolution of singularities.
• To clarify technical concepts such as transforms of ideals, maximal contact, and residual order through concrete examples.
• To investigate the limitations of maximal contact in positive characteristic and the behavior of differential operators in this setting.
• To support graduate-level teaching by including detailed examples and exercises with hints.

Proposed method

• Uses a lecture-based, dictionary-style format to present core concepts in resolution of singularities, with repeated definitions for self-contained access.
• Applies blowup techniques to analyze singularities of algebraic varieties, particularly focusing on ideal transforms and strict transforms.
• Employs local invariants such as order and residual order to track singularities through successive blowups.
• Utilizes differential operators in positive characteristic, especially $\partial_{x_i^{p^k}}$, to study the behavior of polynomials under reduction.
• Analyzes specific examples, including the Camelia surface and hypersurfaces with non-maximal contact, to illustrate theoretical results.
• Relies on computational algebra and coordinate transformations to verify orders and residual orders in blowup charts.

Experimental results

Research questions

• RQ1 Under what conditions does a hypersurface have maximal contact with a given singular variety in positive characteristic?
• RQ2 Why does the Camelia surface fail to admit any regular local hypersurface with maximal contact at the origin?
• RQ3 How do differential operators like $\partial_{x_i^{p^k}}$ behave in positive characteristic, and what do they reveal about polynomial orders?
• RQ4 What is the residual order of a strict transform after blowup, and how does it change under coordinate changes?
• RQ5 Can the top locus of a singular variety be contained in a regular local hypersurface, and what constraints does this impose?

Key findings

• The Camelia surface defined by $f = 27x^2y^3z^2 + (x^2 + y^3 - z^2)^3$ has a singular locus with six irreducible components, including cuspidal curves and complex-conjugate pairs.
• No regular local hypersurface at the origin of the Camelia surface can contain the top locus, which is an irreducible curve parametrized by $t \mapsto (t^{32}, t^7, t^{19}, t^{15})$, due to non-integrality of the exponents.
• The residual order of the strict transform of $f$ after blowup in the $x$-chart increases from $3^4$ to $2 \cdot 3^4$, demonstrating order increase under blowup.
• In positive characteristic, the differential operator $\partial_{x_i^p}$ does not satisfy the Leibniz rule, as $\partial_{x_i^p}(x_i^p) = 1$ while the Leibniz term vanishes.
• For $f = z^{3^5} + x^{4 \cdot 3^4} y^{3^4}(x^{3^4} + y^{3^4} + x^{300})$ in characteristic 3, the residual order increases after blowup, indicating non-trivial singularity persistence.
• The order of the coefficient ideal along the generic point of the $z$-axis is $p$, while the residual order of $f$ is $p+1$, showing a discrepancy in invariants.