By Noam Elkies, Lou M. Pretorius and Konrad J. Swanepoel. Discrete & Computational Geometry 35 (2006), 361–373. Author preprint: arxiv.org/abs/math/0403023

Locator: introduction, real Sylvester-Gallai theorem and the extensions to complex and quaternionic projective spaces.

By Edward Witten. Mathematical Research Letters 1 (1994), 769–796. Author preprint: arxiv.org/abs/hep-th/9411102

Locator: §2, equations, moduli spaces and invariants.

By Ravi Vakil. The title page identifies the November 18, 2017 draft. Author-hosted PDF: math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf

Consulted passages: §0.4 (goals and polynomial equations), Chapter 3 (affine schemes), and the sections on rational maps and birational morphisms.

Projective geometry studies projective spaces and incidence relations such as which points lie on which lines, with transformations induced by invertible linear maps. Over a field $K$, points of ${\mathbb{P}}^n(K)$ are one-dimensional vector subspaces of $K^{n+1}$, represented by nonzero homogeneous coordinates $[x_0:\cdots :x_n]$ up to a common nonzero scalar.

The chart $x_0\ne 0$ gives affine coordinates $(x_1/x_0,\dots ,x_n/x_0)$. Points with $x_0=0$ form a hyperplane at infinity. In the projective plane, distinct lines meet in one point: parallel affine lines acquire their shared direction point at infinity. This statement concerns the projective plane, not arbitrary pairs of lines in higher dimension.

A projective transformation preserves incidence and the cross-ratio, but generally not Euclidean lengths or angles. This makes the subject natural for perspective and for homogeneous equations in algebraic geometry.

Population genomics studies variation across many genomic regions in sampled biological populations. It connects genome measurements to ancestry, gene flow, demographic history and natural selection. Whereas one genetic locus gives one partly stochastic history, many loci provide repeated but potentially correlated evidence. ^[ref]

For example, comparing allele frequencies among populations can reveal broad ancestry patterns while highlighting unusually differentiated regions. Such an outlier is a candidate for explanation, not proof of local adaptation: sampling, linked variation and demographic events also affect the comparison. A sampling design must specify who was included, and reference-dependent sequencing errors can imitate biological differences if they differ systematically between groups.

Reassortment is the mixing of intact genetic segments from different parental viruses into progeny with a segmented genome. If parental viruses reproduce in the same cell, a progeny genome can inherit some segments from one parent and others from the other. This exchanges whole segment identities; it does not require a crossover within each RNA or DNA segment.

For a toy virus with three distinguishable segments, parents $A_1{A}_2{A}_3$ and $B_1{B}_2{B}_3$ allow the reassortant combination $A_1{B}_2{A}_3$. Independent binary choice at each position gives $2^3=8$ possible combinations, including the two parental combinations. This is a combinatorial count; packaging compatibility and viability can prevent many combinations from appearing.

In a 1979 study, RNA oligonucleotide fingerprints and protein peptide maps suggested that an influenza strain retained some segments from earlier H1N1 viruses while other segments most likely came from an H3N2 parent. The unchanged H1N1 surface subtype therefore did not imply unchanged ancestry across the whole genome.^[ref]

A clock gene contributes to a biological timing mechanism, for example the approximately daily circadian rhythm. Its product can participate in a feedback loop whose delayed activity changes the later production or action of other clock components. The term can describe a functional class; it should not be confused with the particular gene named CLOCK.

In the foundational fly experiment, three mutations assigned to the same locus produced arrhythmic behaviour or approximately 19-hour and 28-hour rhythms. The investigators measured both emergence timing in populations and locomotor activity in individuals. This linked inherited variation to an internal timing mechanism, rather than simply to an immediate response to light. ^[ref]

Teosinte branched1, conventionally written tb1 for the gene, encodes a transcription factor that suppresses outgrowth of lateral buds in maize. It is a major component of the difference in apical dominance between cultivated maize and its wild teosinte relatives.

John Doebley and colleagues connected tb1 expression and mutant morphology to suppression of axillary growth. Cultivated maize showed increased tb1 expression in the relevant comparison, linking altered regulation of a developmental gene to a conspicuous domestication trait.^[ref]

This illustrates why regulatory change matters: a large difference in branching need not require inventing an entirely new protein. Changing where or how strongly an existing regulator acts can reshape the plant.

The NDT80 gene of budding yeast encodes Ndt80, a transcription factor involved in progression through meiosis. The gene is the DNA information; Ndt80 is the resulting regulatory protein. Keeping these distinct makes an intervention precise: deleting the NDT80 gene changes production of Ndt80, whereas measuring Ndt80 binding tests a property of the protein.

The original 1995 characterization found that NDT80 gene mutants arrested at pachytene with persistent synaptonemal complexes.^[ref] Subsequent experiments showed that Ndt80 bound upstream DNA and activated transcription of genes involved in meiotic division and reproductive cell formation.^[ref]

SHATTERPROOF1 is an Arabidopsis thaliana gene encoding a MADS-box transcription factor involved in fruit dehiscence. The gene's product helps specify the dehiscence zone, the specialized region along which a mature fruit opens.^[ref]

Its relationship with SHATTERPROOF2 illustrates compensation between related genes. In the reported experiment, the single mutant did not show the novel opening defect seen when both functions were disrupted. Consequently, a normal-looking single mutant does not establish that SHATTERPROOF1 has no developmental role; the appropriate comparison includes the double mutant.^[ref]

SHATTERPROOF2 is an Arabidopsis thaliana gene encoding a MADS-box transcription factor that acts with SHATTERPROOF1 in fruit dehiscence. The 2000 study connected their combined activity to dehiscence zone differentiation and lignin deposition in adjacent cells.^[ref]

Loss-of-function and constitutive-expression experiments ask complementary questions: what fails when the normal regulators are absent, and what changes when their activity is supplied outside its usual pattern? Neither question identifies the transcription factor's direct DNA targets by itself. Establishing regulation of tissue development is a different evidential step from identifying each directly bound gene.

The SHATTERPROOF genes are the related Arabidopsis thaliana genes SHATTERPROOF1 and SHATTERPROOF2, encoding MADS-box transcription factors involved in fruit dehiscence. Their developmental role connects regulation of gene expression to formation of the dehiscence zone and adjacent lignin deposition.^[ref]

The paired experiment matters. Compare the normal plant, each single mutant, and the double mutant. An effect appearing chiefly in the last comparison supplies evidence for genetic redundancy; observing only the normal plant and one single mutant could miss the shared function.^[ref]

The family name groups the two loci; it does not identify one fused gene. The opening phenotype also does not by itself enumerate the transcription factors' direct DNA targets. Distinguishing locus identity, regulatory activity and tissue-level outcome makes the experimental claim precise.

GRIK2 is a human gene encoding the GluK2 subunit of kainate receptors. Older experimental literature commonly calls this subunit GluR6. The symbol refers to the gene; the encoded subunit assembles with other subunits into a ligand-gated ion channel.^[ref]

The subunit is subject to RNA editing, and alternatively processed transcripts can encode different isoforms. Thus locating one gene does not establish that every cell makes one unchanging protein sequence.^[ref]

A motivating 1998 experiment removed the corresponding GluR6 subunit in mice and recorded currents at chemical synapses. Train-evoked mossy-fibre kainate receptor currents were lost, alongside altered responses to kainate.^[ref] Genetic perturbation and direct current measurements connect the subunit to a circuit response more specifically than observing its RNA alone. They do not establish that every human variant has the same consequence as deleting the mouse gene.

GUK1 is the symbol for a gene encoding guanylate kinase. In the human annotation, GUK1 is the protein-coding gene named guanylate kinase 1, identified by NCBI GeneID2987. The gene is a DNA locus; the encoded enzyme catalyses chemistry after its expression.^[ref]

The enzyme transfers a phosphoryl group between ATP and guanosine monophosphate, giving ADP and guanosine diphosphate. A compact biochemical notation isPhosphorus bookkeeping gives $3+1=2+2$. The equation describes the protein's activity, not a chemical reaction performed directly by the DNA.

A 1992 study cloned and characterized the yeast gene for guanylate kinase, connecting an inherited sequence with a measured enzyme activity.^[ref] The organism must be stated when identifying a gene: the same functional symbol in yeast and humans does not mean that both organisms carry one identical DNA sequence. Likewise, a change in transcript abundance is not by itself a measurement of catalytic rate.

Phtheirospermum japonicum YUC3 is a gene encoding a YUCCA flavin monooxygenase involved in auxin biosynthesis in the parasitic plant Phtheirospermum japonicum. Its role connects local gene expression with development of a haustorium, the attachment and invasion organ used in interactions with a host plant.^[ref]

The2016 study localized Phtheirospermum japonicum YUC3 expression to epidermal cells near the host-contact site during early haustorium formation. The pattern coincided with the DR5 auxin response reporter. Reducing Phtheirospermum japonicum YUC3 expression by RNA interference decreased haustorium formation, while expression at other epidermal positions could induce haustorium-like structures.^[ref]

The measurements asked complementary questions. Gene expression localized the candidate mechanism; perturbation tested its developmental contribution; a biochemical assay found approximately threefold greater free indole-3-acetic acid in roots overexpressing Phtheirospermum japonicum YUC3 than in empty-vector controls. A reporter gene pattern alone is not a direct chemical measurement of auxin, and a haustorium-like structure alone does not establish a successful host connection.^[ref]

A 16S ribosomal RNA gene is a DNA sequence transcribed to produce16S ribosomal RNA, a structural and functional RNA of the small ribosomal subunit in bacteria and archaea. It is an RNA-coding gene, rather than a gene whose final product is a protein. The Escherichia coli rrsA annotation supplies a concrete example: NCBI records its gene type as rRNA and its product as16S ribosomal RNA.^[ref]

Sequence comparison makes the 16S ribosomal RNA gene useful as a marker of microbial relationships. The historical motivation includes a1977 analysis by Carl R. Woese and George E. Fox using ribosomal RNA sequence characterization to distinguish deep lineages. That experiment analysed RNA sequence information; it should not be described as a modern polymerase chain reaction survey.^[ref]

Gene copy number complicates abundance interpretation. A2004 genome comparison examined both variable copy numbers and sequence differences between16S copies in one genome.^[ref] In an invented ideal assay,100 cells with one target copy each contribute100 templates, while100 cells with four copies contribute400. Equal cell counts then give20percent and80percent of templates. Extraction, primer matching and amplification can introduce further differences. A sequence-read fraction is consequently not automatically a cell fraction.

The mouse Tlr4 gene is the house mouse gene encoding Toll-like receptor 4. NCBI identifies it as Tlr4, GeneID21898. The DNA locus, its RNA transcripts and its receptor protein are distinct objects: measuring a sequence change does not directly measure receptor abundance or signalling output.^[ref]

A1998 study linked two inherited defects in lipopolysaccharide responsiveness to the mouse Tlr4 gene. In C3H/HeJ mice, the coding change was predicted to replace proline by histidine at receptor position712. C57BL/10ScCr mice carried a mutation eliminating normal receptor function. These animals were relatively resistant to endotoxin while being unusually susceptible to Gram-negative infection.^[ref]

The result motivates a distinction between avoiding an inflammatory response and recognizing an invading bacterium. A lower response to an isolated bacterial component can coexist with impaired defence against infection. It also illustrates the evidence needed to connect genotype and mechanism: association of a mutation with a phenotype, prediction of its protein consequence and measurement of cellular signalling are related steps, rather than interchangeable assays. The1998 host-genetic findings supported a role for Toll-like receptor 4 in lipopolysaccharide recognition and signalling; a coding variant alone is not a direct ligand-binding measurement.

Gene copy number is the count of copies of a specified gene sequence relative to a stated reference, such as one genome or one biological cell. The reference matters: the count per cell can change with the number of genome equivalents even when the count per genome is unchanged. A DNA copy count also differs from gene expression or the amount of its protein product.

A primary example is the 16S ribosomal RNA gene. A2004 study compared gene copies and their sequence differences within bacterial and archaeal genomes. It showed why a marker's copy count and sequence diversity must be considered separately when interpreting community data.^[ref]

For an invented ideal assay,200 cells of population A carry two target copies each, while100 cells of population B carry four each. Both supply400 templates, giving equal template shares despite cell shares of $2/3$ and $1/3$. More generally, a response can scale as $N_i{k}_i{e}_i$, where $N_i$ is cell count, $k_i$ copies per cell and $e_i$ combined recovery/amplification efficiency. Inferring cell numbers then requires information about the other factors. Copy number variation describes a biological source of changing counts; an assay response alone does not establish which source caused a difference.

Drosophila melanogaster rosy, usually written ry, is a protein-coding gene encoding xanthine dehydrogenase, an enzyme involved in purine degradation. FlyBase identifies it as FBgn0003308 and CG7642. The named gene, its enzyme product and the visible eye-colour phenotype used in genetic experiments are different levels of description.^[ref]

The locus became an informative marker in early P-element vector experiments. A 1982 study transferred a wild-type rosy-containing DNA fragment into mutant embryos and obtained animals with inherited chromosomal copies and wild-type eye colour. The phenotype made the introduced sequence's function visible; molecular and inheritance measurements established that more than a transient signal had occurred.^[ref]

As a teaching comparison, two insertions could both restore the marker phenotype while carrying different neighbouring DNA or occupying different genomic positions. The shared visible outcome would not establish identity of the complete constructs. A marker therefore helps select or recognize an outcome, while sequencing, mapping and transmission answer additional questions about what was introduced and where it resides.

HBA1 is the human gene encoding an alpha-globin chain of hemoglobin. It lies in the alpha-globin region of chromosome 16 alongside HBA2. The reference coding sequences of HBA1 and HBA2 are identical, although parts of their surrounding and untranslated sequences differ.^[ref]

The gene is a particular region of DNA; its product is a protein. Two different loci can therefore produce the same protein sequence. An alpha-globin peptide measurement alone need not reveal how much came from HBA1 rather than HBA2. Alpha chains contribute both to hemoglobin A, with two alpha and two beta-globin chains, and to fetal hemoglobin, with alpha and gamma-globin chains.

An original measurement example makes the ambiguity explicit. Let $a_1,a_2$ denote amounts of identical alpha-chain product contributed by the two loci. A perfectly measured total $a=a_1+a_2=100$ units fits contributions $(40,60)$ or $(20,80)$. More precise measurement of the same total cannot distinguish these possibilities. A locus-sensitive RNA or DNA observation supplies a different kind of evidence.

This also separates gene dosage from final hemoglobin abundance. The number of gene copies constrains what genetic material is present, while gene expression, translation and assembly determine how much functional product is produced. Naming HBA1 identifies the locus being discussed, not a unique alpha-chain molecule that can always be traced back to that locus by its amino-acid sequence alone.