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July 22, 2024

The readings during this period are yet to be reflected in Notes on Algebraic Geometry.

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July 22, 2024

2024-07-27 [2024-07-27]
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July 22, 2024

The readings during this period are yet to be reflected in Notes on Algebraic Geometry.

Related

Notes on Algebraic Geometry [ag-0001]

August 13, 2024
Utensil Song

What is Algebraic Geometry? [ag-0005]

Algebraic geometry deals with solution sets of systems of polynomial equations [borisov2024adventures, sec. 1] from an geometric view, with many algebraic concepts imitating the notions of analysis and topology [kriz2021introduction, p. vii], but has the advantage of being able to deal with singularities [mehrle2017algebraic, lect. 1, sec. 1] and other pathological objects.

Algebraic geometry has developed in waves, each with its own language and point of view, see [hartshorne1977graduate, p. xiv] for some discussion. The older language is closer to the geometric intuition, while the newer language developes powerful techniques to solve problems in great generality, but the study of the latter is considered tedious, or even depressing, by many, when not accompanied with tangible applications [grothendieck1964elements, p. 12].

Remark. the grand plan [ag-0002]

We will use [kriz2021introduction] as a holistic guide to the organization of materials, which has done a great job of having the minimal prerequisites, being self-contained, and covering most of the topics that concern us, including the geometric motivation.

For a similar purpose, we use the formalization papers [bordg2022simple] and [buzzard2022schemes] to guide the path, at least the part towards schemes, and their counterparts in the Mathlib of Lean 4.

For prerequisites in basic algebra and commutative algebra, we will use [knapp2006basic] and [knapp2007advanced], and notes by Andreas Gathmann [gathmann2023plane][gathmann2013commutative] and David Mehrle [mehrle2015commutative]. For the geometric motivation and intuition, we will use [cox1997ideals] and [borisov2024adventures].

For upstream treatment of algebraic geometry, we will use [grothendieck1964elements] (particularly the English translation available at ryankeleti/ega) and [fantechi2006fundamental]. For modern notes, we will use [vakil2024rising], [gathmann2022algebraic] and [mehrle2017algebraic], with an eye on the classic textbook [hartshorne1977graduate].

We also need to tap into the language of Stacks in a modern setting, as treated in [khan2023lectures], with preliminaries on \(\infty \)-categories and derived categories.

See the plan for notes on algebraic geometry for an early discussion of the plan.

Convention. rings and fields [gathmann2013commutative, 0.1] [ag-0008]

A ring, usually denoted \(R\), is always assumed to be a commutative ring (i.e. \(a+b=b+a\) and \(a b=b a\) for all \( a, b \in R\)) with 1 (i.e. the multiplicative identity element, or called multiplicative unit).

\(1 \neq 0\) is not required, where \(0\) is the additive neutral element. If \(1=0\), then \(R\) must be the zero ring (or called the trivial ring), which consisting of one element, and is denoted \(\{0\}\).

Subrings must have the same unit, and ring homomorphisms are always required to map \(1\) to \(1\).

A field, usually denoted \(K\), is a commutative ring with \(1\), where every nonzero element has a multiplicative inverse (thus division can be defined).

Definition. formal variable, formal expression [ag-0009]

A formal variable is an arbitrary symbol that is used to represent some mathematical object, and assumes nothing about the value or nature of the object.

A formal expression is mathematical expression with formal variables, assuming nothing but the formal variables participate in the expression have operations that are used in the expression.

A formal expression can be evaluated by replacing the formal variables with actual mathematical objects that have the operations defined in the expression.

When there is no ambiguity, we may omit the word "formal" and simply say variables or an expression.

Definition. monomial [cox1997ideals, 1.1.1] [ag-0003]

A monomial in \(n\) formal variables \(x_1, \ldots , x_n\), denoted \(x^\alpha \), is a formal expression of the form \[ x_1^{\alpha _1} x_2^{\alpha _2} \cdots x_n^{\alpha _n} \] where \(n \in \mathbb N\), \(\alpha =\left (\alpha _1, \ldots , \alpha _n\right )\) is an \(n\)-tuple of nonnegative integers.

The total degree of the monomial is denoted \(|\alpha |=\alpha _1+\cdots +\alpha _n\).

Definition. polynomial [cox1997ideals, 1.1.2, 1.1.3] [ag-0004]

A polynomial \(f\) over a ring \(R\) in \(n\) variables is a finite linear combination (with coefficients \(a_\alpha \) in \(R\) ) of monomials, i.e. the formal expression of the form \[ f=\sum _\alpha a_\alpha x^\alpha , \quad a_\alpha \in R, \]

The set of all polynomials in \(x_1, \ldots , x_n\) with coefficients in \(R\) is denoted \(R\left [x_1, \ldots , x_n\right ]\).

\(a_\alpha x^\alpha \) is called a term of \(f\) if \(a_\alpha \neq 0\).

The total degree of \(f \neq 0\), denoted \(\operatorname {deg}(f)\), is the maximum \(|\alpha |\) such that the coefficient \(a_\alpha \) is nonzero. The total degree of the zero polynomial is undefined.

Remark. polynomial ring [cox1997ideals, 1.1.3] [ag-0007]

Under addition and multiplication, \(R\left [x_1, \ldots , x_n\right ]\) satisfies all axioms of a commutative ring, and for this reason we will refer to \(R\left [x_1, \ldots , x_n\right ]\) as a polynomial ring.

Definition. affine space [gathmann2013commutative, 0.3] [ag-000A]

The (\(n\)-dimensional) affine space over a field \(K\), denoted \(\mathbb {A}_K^n\), is \[ \left \{\left (c_1, \ldots , c_n\right ): c_i \in K \text { for } i=1, \ldots , n\right \} \] which is just \(K^n\) as a set, without the its additional structures as a \(K\)-vector space and a ring.

We'll often use the term affine \(n\)-space to indicate the dimension. Particularly, an affine 1-space is called an affine line, an affine 2-space is an affine plane.

Definition. affine variety [gathmann2013commutative, 0.3] [ag-000D]

Let \(S \subset K\left [x_1, \ldots , x_n\right ]\) be a set of polynomials. The zero locus (or zero set) of \(S\) is \[ V(S):=\left \{x \in \mathbb {A}_K^n: f(x)=0 \text { for all } f \in S\right \} \subset \mathbb {A}_K^n \]

An affine algebraic variety over \(K\) is a subset of \(\mathbb {A}_K^n\) of this form. It's usually simply called an affine variety over \(K\), or an affine \(K\)-variety.

If \(S=\left (f_1, \ldots , f_k\right )\) is a finite set, \(V(S)\) can be written as \(V\left (f_1, \ldots , f_k\right )\).

Obviously, it is the set of all solutions of the system of polynomial equations \(f_1\left (x_1, \ldots , x_n\right )=\cdots =f_s\left (x_1, \ldots , x_n\right )=0\) [cox1997ideals, 1.2.1], denoted \(\operatorname {Sol}(S;K)\) [dolgachev2013introduction, p. 1].

Remark. affine v.s. projective [michalek2021invitation, ch. 2] [ag-000E]

The prefix "affine" of affine variety is used to distinguish it from a projective variety. Affine varieties arise from arbitrary polynomials, while projective varieties arise from systems of homogeneous polynomials, i.e. linear combinations of monomials of fixed degree.

Since affine varieties are the general case, they are sometimes simply called varieties.

Figure out its relation to affine space in Geometry, which preserves parallelism and ratio of lengths for parallel line segments, but not distances and measures of angles.

Example. varieties [ag-000F]

\(\mathbf {V}\left (10 x^2-x^3-y^2\right )\) from [cox1997ideals, p. 24]:

$\text{[math]}$

\(\mathbf {V}\left (x^2-y^2 z^2+z^3\right )\) from [cox1997ideals, p. 7, p. 16]:

$\text{[math]}$

Figure. Implicit surfaces [uts-000J]

#define AS_LIB 1

int get_shape() {
    return int(iTime) % 52;
}

#include "/forest/shader/implicit.glsl"

void mainImage( out vec4 fragColor, in vec2 fragCoord )
{
    vec2 uv = 2.*(fragCoord-iResolution.xy/2.)/iResolution.y; // contains [-1,1]^2
    vec3 col = vec3(0.);
    
    // Camera rays
    vec3 camPos = vec3(4.,0.,0.);
    vec3 camDir = - normalize(camPos);
    vec3 rayPos, rayDir;
    float zoom = 1.3; // 1.8*cos(iTime);
    // if (checkKey(KEY_E)) zoom = 0.5;
    float fov = 0.4*zoom;
    float fov_ortho = 1.5*zoom;
    #if perspective
        // perspective cam
        rayPos = camPos;
        rayDir = normalize(camDir + fov*vec3(0., uv.x, uv.y));
    #else
        // orthographic cam
        rayPos = camPos + fov_ortho*vec3(0., uv.x, uv.y);
        rayDir = camDir;
    #endif
    // for perspective background in orthographic mode
    vec3 cubemapDir = normalize(camDir + fov*vec3(0., uv.x, uv.y));
    
    // Mouse-controlled rotation
    vec2 mouse = initMouse + vec2(0.015625*sin(iTime*PI), 0.0); // initMouse; // iMouse.xy == vec2(0.,0.) ? initMouse : (iMouse.xy/iResolution.xy - 0.5);
    float yaw = clamp(- mouse.x * 2.*PI * 1., -PI,PI);
    float pitch = clamp( mouse.y * PI * 1.2, -PI*0.5, PI*0.5);
    // pitch and yaw rotations (column-wise matrices)
    mat3 rot = mat3(cos(yaw), sin(yaw), 0., -sin(yaw), cos(yaw), 0., 0., 0., 1.);
    rot = rot * mat3(cos(pitch), 0., -sin(pitch), 0., 1., 0., sin(pitch), 0., cos(pitch));
    // apply
    camPos = rot*camPos;
    camDir = rot*camDir;
    rayPos = rot*rayPos;
    rayDir = rot*rayDir;
    cubemapDir = rot*cubemapDir;
    
    //cubemapDir = vec3(cubemapDir.x, cubemapDir.z, cubemapDir.y);
    
    vec3 hitPoint = raycast(rayPos, rayDir);
    
    if (hitPoint == BINGO) { fragColor = vec4(BINGO,1.0); return; }
    //if (hitPoint == NOHIT) { fragColor = vec4(NOHIT,1.0); return; }
    //if (hitPoint == NOBOUNDHIT) { fragColor = vec4(NOBOUNDHIT,1.0); return; }
    //if (hitPoint == ESCAPEDBOUNDS) { fragColor = vec4(ESCAPEDBOUNDS,1.0); return; }
    //if (hitPoint == MAXDISTREACHED) { fragColor = vec4(MAXDISTREACHED,1.0); return; }
    //if (hitPoint == MAXITERREACHED) { fragColor = vec4(MAXITERREACHED,1.0); return; }
    
    if (hitPoint == NOBOUNDHIT  || hitPoint == NOHIT || hitPoint == ESCAPEDBOUNDS || hitPoint == MAXITERREACHED) {
        //fragColor = vec4(vec3(0.2),1.0); return;

        // make background transparent
        fragColor = vec4(0.0,0.0,0.0,0.0); return;
        
        col = with_background(cubemapDir);
        
        #if showBoundingCube
        // darken bounding cube
        if (hitPoint != NOBOUNDHIT) { col *= vec3(0.7); }
        #endif

        fragColor = vec4(col,1.0); return;
    }

    vec3 grad = gradf(hitPoint+1.1*EPS*(-rayDir));
    float s = -sign(dot(grad,rayDir));
    
    col = with_color_mode(grad, s, hitPoint, camPos);

    col = clamp(col, 0., 1.);

    col = with_surface_pattern(col, hitPoint);

    col = with_shading(col, grad, s, rayDir);
    
    col = clamp(col, 0., 1.);

    fragColor = vec4(col,1.0);
}

adapted from Rogolop's Implicit surfaces

appendix [ag-000B]

For draft notes, see Drafts for Notes on Algebraic Geometry.