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, 144, 146, 147 dependent, 149, 151, 152 disjoint, 146–147 explanation of, 129 independent, 149, 151–153 mutually exclusive, 129, 146–147, 153 relationships between, 144–148 simple, 129–134 Exhaustive events, 158–159 Expected value for continuous random variables, 170 of random variable, 166–167 Experimental design completely randomized, 450–458 elements of, 448–449 explanation of, 4–5, 255–258, 448–449 randomized block, 466–473 two-way classiﬁcation, 478–484 Experimental error, 466, 488 Experimental units, 3, 8, 448 Experimentation, 257, 448 Experiments binomial, 184–186, 614 examples of, 128–129 explanation of, 128 factorial, 473, 478–484 multinomial, 595, 610 paired, 644–648 Exponential random variable, 222 Extended mn Rule, 138–139 Extrapolation, 519 Factorial experiments explanation of, 473, 480 tests for, 482 2 3, 480 a b, 478–484 Factorial notation, 139 Factors, in experiments, 448 False negatives, 160 False positives, 160 F distribution analysis of variance and, 454 explanation of, 425–430 percentage points of, 694–701 First-order models, 567 Fitted line, 527–531 Five-number summary box plots created by, 81–82 explanation of, 80 F probabilities applet, 456, 470 explanation of, 425–427 Frequencies, 11, 12, 99 Frequency histograms, 24. See also Relative frequency histograms Friedman, Milton, 656 Friedman Fr-test for randomized block designs, 656–659 INDEX ❍ 739 F table, 426 F-test analysis of, 518 analysis of variance, 481–482, 518, 556 for comparing population means, 455, 464 explanation of, 423, 429, 430, 432 Inference. See also Small-sample inference applications for, 298, 299 Central Limit Theorem and, 266 concerning population variance, 417–423 concerning slope of line, 514–516 goodness of, 299 independent random samples and, General linear model, 552–553 General Multiplication Rule, 149–150 Goodness-of-ﬁt test, 597–599, 615 Goodness of the inference, 299 Gosset, W
. S., 388 Graphs. See also Charts; speciﬁc types of graphs for categorical data, 11–14 interpretation of, 22–24 limitations of, 53 of Poisson probability distribution, 200 of probability distributions, 168 for qualitative variables, 98–100 for quantitative data, 17–24 residuals, 488–489 of straight line, 108 using MINITAB, 38–42 Histograms applet, 35–37 probability, 165 relative frequency, 24–29, 35–37 using MINITAB, 41–42, 69 Homogeneity tests, 611, 614 Hypergeometric probability distribution, 205–207 Hypothesis testing binomial proportions, 368–371 conﬁdence intervals and, 365–366 considerations regarding, 378 correlation coefficient, 536–537 difference between two binomial proportions, 373–376 399–406 methods for, 5, 298–299 paired-difference test and, 410–414 population mean and, 391–396 reliability of, 5 Inferential statistics explanation of, 4 steps in, 4–5 Interaction plots, 479 Interaction term, 567 Intercept, 553 Interquartile range (IQR), 78–79 Intersection, of events, 144–146 Interval estimate, 300 Interval estimation construction of conﬁdence interval and, 308–309 explanation of, 300, 307 interpretation of conﬁdence interval and, 311–314 large-sample conﬁdence interval and, 310–311, 314–315 Interval estimators, 300, 307 Judgment samples, 258 Kendall, 660 Kruskal-Wallis H-test for completely randomized designs, 650–654 Large-sample conﬁdence interval explanation of, 310–311, 319–320 for population proportion, 314–315 difference between two population means, Large-sample estimation 363–366, 401–402 equality of population variances, 427–428 explanation of, 298–299, 344 one-tailed, 345, 347, 349 paired-difference, 412–414 population mean, 347–360, 391, 392 population parameters, 344 population variance, 420–421 slope of line, 515–516 small-sample, 391–392 statistical, 344–347 two-tailed, 345, 349, 350, 645 Hypothetical populations, 257 choosing sample size and, 329–333 estimating difference between two binomial proportions and, 324
–326 estimating difference between two population means and, 318–321 estimator types and, 299–300 interval estimation and, 307–315 one-sided conﬁdence bounds and, 328–329 point estimation and, 300–305 statistical inference and, 298–299 Large-sample tests of hypotheses about population mean, 347–360 for binomial proportion, 368–371 for difference between population means, Independence, chi-square test of, 363–366 602–603, 605 Independent events explanation of, 149, 151 multiplication rule for, 151–152 mutually exclusive vs., 153 Independent random samples analysis of variance table for, 452 explanation of, 399–406 Wilcoxon rank sum test and, 630–637 Independent variables, 108 Indicator variables, 567 for difference between two binomial proportions, 373–376 Law of Total Probability, 159, 160 Least-squares estimators, 507–508 Least-squares line, 506 Least-squares method applets, 506, 508 explanation of, 506–508, 554–555 Least-squares regression line, 109–110, 506, 508 Left inclusion, method of, 25 740 ❍ INDEX Left-tailed test, 346 Level, in experiments, 448 Line ﬁtted, 527–531 graph of, 108 of means, 505, 514–516, 527, 528, 553 Linear correlation, 537 Linear probabilistic model, 503–506 Linear regression analysis of variance for, 509–511 checking assumptions in, 522–527 correlation analysis and, 533–537 ﬁtted line and, 527–531 MINITAB, 510–511, 530–531, 540–543 testing usefulness of, 514–520 Line charts, 19 Location, of data distribution, 22, 24 Log-linear models, 616 Lower conﬁdence limit (LCL), 309, 328 Lower quartiles, 77, 78 Main effect sums of squares, 480 Mann-Whitney test, 634 Mann-Whitney U test, 630 Marginal probabilities, 603 Margin of error estimation of, 303–305 explanation of, 302 Means for binomial random variables, 186–188 calculating probabilities for, 268–272 correction for, 451 for discrete random variable, 166–170 explanation of, 54 for grouped data, 74 line of, 505, 5
14–516, 527, 528, 553 for Poisson probability distribution, 198 population, 54, 166, 303, 304, 310–311, 318–321, 347–360 sample, 54, 55, 266–272 small-sample inferences for difference between two, 410–414 standard error of, 267, 392 use of, 57 Mean squares in analysis of variance, 452, 454, 457, 468, 470, 510 explanation of, 452, 454, 468 Measurement, 8 Measures. See Numerical measures Measures of center applet, 56 explanation of, 53 mean as, 54–56 median as, 55–56 mode as, 57 Median explanation of, 55–56 use of, 57 Method of least squares. See Least-squares method Method of left inclusion, 25 MINITAB analysis of variance, 453–454, 458, 469, 483, 489–490, 492–495, 510, 653 bar charts, 13–14, 39, 40 bivariate data, 115–118 box plots, 83 Central Limit Theorem, 288–290 chi-square test, 599, 617–620 correlation coefficient, 107, 108 cumulative distribution function, 192 discrete probability distributions, 173–175 dotplots, 20, 23, 40, 41 ﬁtted quadratic regression line, 561 F-test, 423, 429 goodness-of-ﬁt test, 599 graphs, 38–42 histograms, 41–42, 69 introduction to, 37–38 Kruskal-Wallis H-test, 653 linear regression, 510–511, 530–531, 540–543 Mann-Whitney test, 634 nonparametric statistical methods, 668–671 normal probabilities, 246–248 numerical measures, 80, 88–89 paired t, 414 pie charts, 38–40 Poisson and binomial probabilities, 202, 209–211 pooled method, 406, 410 probability plots, 489 p-values, 354, 395, 557 regression, 517, 530, 562, 569–571, 577, 583–585 relative frequency histogram, 27 sample quartiles, 80 sampling distributions, 264 Satterthwaite approximation, 406 small-sample testing and estimation, 423, 434–436 t-test, 557 Tukey’s test, 465 use of, 6 variance, 534 Wilco
xon rank sum test, 634 Wilcoxon signed-rank test, 647 x– chart, 283 mn Rule application of, 142 explanation of, 137–138 Extended, 138–139 Modal class, 57 Mode, 57 Monte Carlo procedure, 295 Multicollinearity, 580–581 Multidimensional contingency tables, 616 Multinomial experiments, 595, 610 Multinomial populations, 610–612 Multiple R2, 557 Multiple regression analysis analysis of variance for, 555–556 checking regression assumptions and, 558–559 estimation and prediction and, 559 explanation of, 552–554 general linear model for, 552–553 interpreting results of signiﬁcant regression and, 557–558 least-squares method for, 554–555 MINITAB, 569–571, 577, 583–585 misinterpretation of, 580–581 polynomial regression model and, 559–562 quantitative and qualitative predictor variables and, 566–571 residual plots and, 578–579 steps in, 582 stepwise regression analysis and, 579–580 testing sets of regression coefficients and, 575–577 usefulness of regression model and, 556–557 Multiplication Rule general, 149–150 for independent events, 151–152 Law of Total Probability and, 159 use of, 150, 154 Multivariate data, 9 Mutually exclusive events exhaustive and, 158–159 explanation of, 129, 146–147 independent vs., 153 Nonparametric statistical methods analysis of variance and, 490 explanation of, 630 Friedman Fr-test for randomized block designs as, 656–659 Kruskal-Wallis H-test for completely randomized designs as, 650–654 MINITAB, 668–671 rank correlation coefficient as, 660–664 sign test for paired experiment as, 639–641 statistical test comparisons and, 643–644 Wilcoxon rank sum test as, 630–637 Wilcoxon signed-rank test for paired experiment as, 644–648 Nonparametric testing, 490 Nonresponse, in sample surveys, 256 Normal approximation to binomial probability distribution, 237–243 for sign test, 640–641 for Wilcoxon rank sum test, 634–637 for Wilcoxon signed-rank test, 647–648 Normal curves areas under, 226–228, 688–689 binomial probabilities and, 238 calculating probabilities under, 228, 229, 231, 232
mutations counting rule for, 140, 141 explanation of, 139 Pie charts explanation of, 12, 13 for quantitative data, 17–19 side-by-side, 98–100 using MINITAB, 38–40 Plane, 553 Plot of residual versus ﬁt, 523, 558 Point estimate, 299 Point estimation explanation of, 299 large-sample, 325–326 of population parameter, 302–303 use of, 300–305 Point estimator explanation of, 299, 315 sampling distribution of, 301 variability of, 303 Poisson approximation, to binomial distribution, 201–202, 237 Poisson probabilities, 237 Poisson probability distribution explanation of, 197–202 formula for, 198 graphs of, 200 MINITAB, 202, 209–211 Poisson random variable, 197–198 Polls, 1–3 Polynomial regression models, 560–562 Pooled method, 403, 405, 406 Pooled t test, 410 Population correlation coefficients, 536 Population means estimating difference between two, 318–321 estimation of, 303, 304, 318–321 explanation of, 54, 166 F test for comparing, 455 large-sample conﬁdence interval for, 310–311 large-sample test about, 347–360 large-sample test for difference between, 363–366 ranking, 462–465 small-sample inferences concerning, 391–392 small-sample inferences for differences between two, 399–406 use of, 55 Population model, 503–506 Population rank correlation coefficient, 664 Populations comparing multinomial, 610–612 explanation of, 3, 8 hypothetical, 257 identiﬁcation of, 4 normal, 266 skewed, 266 symmetric, 266 Population standard deviation, 167, 223 Population variances comparing two, 424–430 estimation of, 64 explanation of, 62 formula for, 167 hypothesis testing of, 420–421 inference concerning, 417–423 inferences concerning, 417–423 Posterior probabilities, 160, 161 Power of statistical test, 357–360, 643 of z-test, 360 Power curve, 357 INDEX ❍ 741 Prediction use of ﬁtted line for, 527–531 use of regression model for, 559 value of, 520 Prediction equation, 503 Prediction intervals, 529 Predictor variables explanation of, 504, 552 in regression models, 566–571 Principle of least squares, 506–507 Prior probabilities, 160 Probabilistic model, simple linear,
503–506 Probabilities Bayes’ Rule and, 158–161 binomial, 201 conditional, 149–151, 159–160 counting rules and, 137–142 cumulative binomial, 188–189 event relations and, 144–148 events and sample space and, 128–131 for general normal random variables, 229–232 independence and, 149–154 laws of total, 159 Multiplication Rule and, 149–152, 154, 159 Poisson, 198–202 posterior, 160, 161 prior, 160 relationship between statistics and, 128 of sample mean, 268 simple-event, 131–134 unconditional, 159, 603 for unions and complements, 146–148 Probability density function, 192, 221, 222 Probability distributions binomial, 184–193, 237–243 chi-square, 418–420 continuous, 220–222 for continuous random variables, 220–223 for discrete random variables, 164–170 explanation of, 163–164, 221 graphs of, 168 hypergeometric, 205–207 MINITAB and, 173–175 normal, 68, 223–232 Poisson, 197–202 requirements for discrete, 164–165 Probability histograms, 165 Probability tables, 131, 148, 159 Process mean, 281–283 Proportions of defectives, 283–285 estimating difference between two binomial, 324–326 estimation of, 303 sample, 275–279 pth percentile, 76 p-value calculation of, 351–355, 395 explanation of, 345, 346 hypothesis tests and, 365, 378, 394, 422 Quadratic model, 559–562 Qualitative variables explanation of, 10–11, 163 graphs for, 98–100 predictor, 566–571 statistical tables for, 11–12 742 ❍ INDEX Quantitative data graphs for, 17–24 numerical measures for bivariate, 105–107 Quantitative variables explanation of, 10–11, 17, 163 graphs of, 19 predictor, 566–571 scatterplots for two, 102–104 Quartiles calculation of sample, 78–80 explanation of, 78 lower, 77, 78 upper, 77, 78 Quota samples, 258 R2 adjusted value of, 557–558 explanation of, 556–557 Random error, 504, 505 Randomized assignment, 451 Randomized block designs analysis of variance for, 467–473 cautions regarding, 473 explanation of, 413–414, 466–467 Friedman Fr-test for, 656–659
tests for, 471 Random numbers, 256 Random numbers table, 706–707 Random samples explanation of, 256 independent, 399–406, 630–637 1-in-k systematic, 258 simple, 255–256 stratiﬁed, 257–258 Random variables binomial, 184, 186–188, 237, 275 continuous, 163, 170, 220–223 discrete, 163–170, 221 explanation of, 163 exponential, 222 hypergeometric, 205–206 normal, 225–232, 266 Poisson, 197–198 probability density function for, 221 uniform, 222 Range approximation of, 70–71 explanation of, 60–61 interquartile, 78–79 Rank correlation coefficient, 660–664 Rank sum, 631, 644 Regression, 109. See also Linear regression; Multiple regression analysis Regression analysis computer software for, 517 misinterpretation of, 580–581 predicting value of, 522 stepwise, 579–580 Regression coefficients partial, 575–577 testing sets of, 575–577 Regression line. See also Least-squares regression line calculation of, 111–112 explanation of, 109, 506 Rejection region, 346, 347, 349, 350, 352, 355, 393 Relative efficiency, 644 Relative frequencies explanation of, 11, 12, 100 sum of, 221 Relative frequency distributions for increasingly large sample sizes, 220 probability distributions and, 166 showing extreme values on mean and median, 56 Relative frequency histograms construction of, 26–28, 35–37 explanation of, 24–25 uses for, 28–29 using MINITAB, 27 Relative standing measures explanation of, 75 MINITAB and, 80 sample quartiles and, 78 types of, 75–78 Replications, of experiment, 480 Residual, 488 Residual error, 511, 523 Residual plots explanation of, 488–490, 523–524, 558 interpretation of, 578–579 Response, in experiments, 448 Response variables, 504, 552 Right-tailed test explanation of, 346, 454 use of, 596 Robust, 391, 433, 449 s, calculation of, 70–71 s2 calculation of, 63, 64, 401, 418 explanation of, 62 number of degrees of freedom (df ) associated with, 388, 400 Sample mean calculating probabilities for, 268 formula for, 54 sampling distribution of, 266–272 use of, 55 Sample proportion
calculating probabilities for, 277–279 sampling distribution for, 275–279 Samples cluster, 258 convenience, 258 elements of, 3 explanation of, 3, 8, 55 judgment, 258 quote, 258 selection of, 255, 256 variance of, 62 Sample size. See also Large-sample conﬁdence interval; Large-sample estimation; Large-sample tests of hypotheses; Small-sample inference; Small-sample techniques binomial experiments and, 186 Central Limit Theorem and, 266 choice of, 329–333 formulas to determine, 333 margin of error and, 305 Sample space, 129, 130 Sample surveys objectives of, 314 problems related to, 256–257, 315 Sample variance calculation of, 64 explanation of, 62–63 Sample z-scores, 75 Sampling, 3 Sampling design. See Experimental design Sampling distributions Central Limit Theorem and, 263–266 MINITAB, 264 of point estimator, 301 of sample mean, 266–272, 318–321 of sample proportion, 275–279 sampling plans and experimental designs and, 255–258 statistical process control method and, 281–285 statistics and, 260–262 Sampling error, 305 Sampling plans, 255–258, 329, 448. See also Experimental design Sampling procedure, 4–5 Satterthwaite’s approximation, 406 Scales, examination of, 22, 24 Scatterplots applets, 104, 504, 505 explanation of, 102 to show correlation, 535–536 for two quantitative variables, 102–104 Screening tests, 159–160 Second-order models, 560, 567 Sequential sums of squares, 556 Shape, of data distribution, 22, 24 Shared information, 581 Shortcut method for calculating s2, 63 Side-by-side pie charts, 98–100 Sigma (), 54 Signiﬁcance level explanation of, 347, 348, 352, 356 practical importance and, 370–371 Sign test normal approximation for, 640–641 for paired experiment, 639–640 Simple events applet, 138 explanation of, 129–131 probabilities of, 131–134, 164 Simple linear probabilistic model, 503–506 Simple random samples, 255–256. See also Random samples Simulation to approximate discrete probability distributions, 165 to approximate sampling distributions, 260–261, 265 Monte Carlo procedure and, 295 Skewed distributions, 22–23, 56 Slope conﬁdence interval for, 517 explanation of, 108 of line
of means, 514–516 partial, 553 test for, 516 Small-sample inference. See also Inference concerning population mean, 391–396 concerning population variance, 417–423 independent random samples and, 399–406 paired-difference test and, 410–414 Small-sample techniques assumptions of, 432–433 comparing two population variances, 424–430 explanation of, 387 MINITAB, 434–436 Student’s t distribution, 387–391 use of, 630 Sources of variation, 468, 481 Spearman rs, 660 Spearman’s rank correlation coefficient critical values of, 705 explanation of, 660–664 Spearman’s rank correlation test, 663–664 Stacked bar charts explanation of, 98 use of, 99, 100 Standard deviation for binomial random variables, 186–188 calculation of, 65 for discrete random variables, 166–170 explanation of, 62–63 for Poisson probability distribution, 198 population, 167, 223 practical signiﬁcance of, 66–70 in research results, 304 Standard error of estimator, 267, 528 explanation of, 313 of mean, 267, 392 in research results, 304 Standardized normal distribution, 225, 229–230 Standardized test statistic, 348 Standard normal random variable, 225–229 Standard normal z distribution, 388 States of nature, 160 Statistical inference. See Inference Statistical process control (SPC) control chart for process mean and, 281–283 control chart for proportion defective and, 283–285 explanation of, 281 Statistical signiﬁcance, 352, 370–371 Statistical software, 6 Statistical tables, 11, 12, 14 Statistical tests. See also Hypothesis testing comparison of, 643–644 equivalence of, 614–615 essentials of, 348–350 explanation of, 344–347, 378 large-sample, 350–351, 363–366, 369–370, 373–376 left-tailed, 346 power curve for, 357 power of, 356–360 right-tailed, 346, 454, 596 Statistical theorems, 261, 262 Statistics descriptive, 4 explanation of, 53, 260 inferential, 4–5 relationship between probability and, 128 sampling distributions and, 260–262 training your brain for, 5–6 Stem and leaf plots, 20–22 Stepwise regression analysis, 579–580 Strata, 257 Stratiﬁed random samples, 257–258
Studentized range explanation of, 463 percentage points of, 708–711 Student’s t distribution applet, 389 assumptions behind, 391 explanation of, 388–389 statistical computing packages and, 395 Student’s t probabilities, 389, 392, 412 Student’s t table, 389 Sum of sample measurements xi, 265 Sum of squares for error (SSE), 452, 480, 506 Sum of squares for treatments (SST), 451–452 Sums of squares calculation of, 481 main effect, 480 sequential, 556 use of, 507, 509 Symbols, for process of summing, 54 Symmetric distributions, 22, 56, 223 Tchebysheff’s Theorem calculation of s and, 70 explanation of, 66–69 use of, 66, 68–70, 193 z-scores and, 76 Tests of homogeneity, 611 Test statistic analysis of variance, 454, 455 explanation of, 345, 346, 405 modiﬁcation of, 375–376 standardized, 348 use of, 349, 393 for Wilcoxon signed-rank test, 644 Tied observations, 639 Time-dependent multinomials, 615–616 Time series, 19, 523 Total sum of squares (TSS), 451, 452 t-probabilities, 537 Treatment means estimating differences in, 456–458, 464 testing equality of, 454–456 Treatments in experiments, 448 identifying differences in, 472–473 randomized block design and, 467 testing equality of, 470–471 Tree diagrams, 130–131 Trend, 19 t statistic degree of freedom for, 400 as robust, 391 use of, 432 t-test paired, 414, 641 pooled, 410 two-sample, 404, 406 use of, 641 Tukey’s method for paired comparisons, 463, 464, 484 Two-sample t test, 404, 406 Two-sided conﬁdence intervals, 328. See also Conﬁdence intervals Two-tailed test of hypothesis, 345, 349, 350, 401, 645 INDEX ❍ 743 2 3 factorial experiments, 480 Type I error, 347 Type II error, 356, 357 Unbiased estimator, 301–303 Unconditional probabilities, 159, 603 Undercoverage, in sample surveys, 257 Uniform random variables, 222 Unimod
al distributions, 23 Unions calculating probabilities for, 146–148 of events, 144–146 Univariate data, 9 Upper conﬁdence limit (UCL), 309, 328 Upper quartiles, 77, 78 Variability estimator, 301–303 measures of, 60–65 rules for describing, 67 Variables continuous, 10, 11, 17 continuous random, 163, 170, 220–223 dependent, 108 discrete, 10, 11, 17 dummy, 567 explanation of, 8–9, 163 independent, 108 predictor, 552 qualitative, 10, 11, 98–100, 163, 566–571 quantitative, 10, 11, 17, 102–104, 566–571 random, 163–170 response, 552 types of, 10–11 Variance. See also Analysis of variance (ANOVA) calculation of, 63 common, 449 explanation of, 62 for grouped data, 74 MINITAB, 534 notation for, 62 population, 62, 64, 167, 417–430 sample, 62, 64 Venn diagrams events in, 146, 147 explanation of, 130 Weighted average, 400 Wilcoxon, Frank, 631, 644 Wilcoxon rank sum test explanation of, 630–634 normal approximation for, 634–636 use of, 637 Wilcoxon signed-rank test critical values of T for, 704 normal approximation for, 647–648 for paired experiment, 644–647 Wording bias, in sample surveys, 257 x– chart, 282–283 y-intercept, 108, 109, 530 z-scores applet, 232–233 explanation of, 75–76 z-test, 358–360 Credits This page constitutes an extension of the copyright page. We have made every effort to trace the ownership of all copyrighted material and to secure permission from copyright holders. In the event of any question arising as to the use of any material, we will be pleased to make the necessary corrections in future printings. Thanks are due to the following authors, publishers, and agents for permission to use the material indicated. Introduction. 1: © Mark Karrass/CORBIS; 2: “Hot News: 98.6 Not Normal,” © McClatchy-Tribune Information Services. All Rights Reserved. Reprinted with permission. Chapter 1. 7: © Jupiterimages/Brand X/CORBIS; 9: Portions of the input and output contained in this publication/book
are printed with permission of Minitab® Inc. All material remains the exclusive property and copyright of Minitab®, Inc. All rights reserved. www.minitab.com; 31, Exercise 1.29: Adapted from “Top Ten Organized Religions of the World,” www.infoplease.com/ipa/A0904108.html, as it appeared on November 15, 2007 Info Please Database, © Pearson Education, Inc. Reproduced by permission of Pearson Education, Inc. publishing as Info Please. All rights reserved; 47, exercise 1.58: Used by permission of GEICO. Chapter 2. 52: © Joe Sohm-VisionsofAmerica/Photodisc/Getty. Chapter 3. 97: © Janis Christie/Photodisc/Getty Images; 126: © 2007 by Consumers Union of U.S., Inc., Yonkers, NY 10703-1057, a nonproﬁt organization. Reprinted with permission from the September 2007 issue of CONSUMER REPORTS® for educational purposes only. No commercial use or reproduction permitted. www.ConsumerReports.org®. Chapter 4. 127: © Tammie Arroyo/Getty Images. Chapter 5. 183: © Kim Steele/Photodisc/Getty Images; 218: From The New York Times, 5/21/1987, p. A22. Copyright © 1987 The New York Times. All rights reserved. Used by permission and protected by the Copyright Laws of the United States. The printing, copying, redistribution, or retransmission of the Material without express written permission is prohibited. Chapter 6. 219: © AFP/Getty Images. Chapter 7. 254: © PictureNet/CORBIS; 291, Exercise 7.66: From Newsweek, Oct. 26, 2006, © 2006 Newsweek, Inc. All rights reserved. Used by permission and protected by the Copyright Laws of the United States. The printing, copying, redistribution, or retransmission of the Material without express written permission is prohibited; 293, exercise 7.78: From J. Hackl, Journal of Quality Technology, April 1991. Used by permission. CREDITS ❍ 745 Chapter 8. 297: © Associated Press; 306, Exercise 8.14: Reprinted with permission from Science News, the weekly newsmagazine of Science, copyright 1989 by Science Services, Inc.; 322, Exercise 8.43: From “Performance Assessment of a
Standards-Based High School Biology Curriculum” by W. Leonard, B. Speziale and J. Pernick in The American Biology Teacher 2001, 63(5); 310–316. Reprinted by permission of National Association of Biology Teachers; 323, Exercise 8.46: From “Performance Assessment of a StandardsBased High School Biology Curriculum” by W. Leonard, B. Speziale and J. Pernick in The American Biology Teacher 2001, 63(5); 310–316. Reprinted by permission of National Association of Biology Teachers; 338, Exercise 8.101: From a CBS/New York Times poll, “Is America Ready For A Woman President?”, Febuary 5, 2006. Copyright © 2006 CBS Broadcasting Inc. All Rights Reserved. Used courtesy of CBS News. Chapter 9. 343: © Scott Olson/Getty Images. Chapter 10. 386: © CORBIS SYGMA; 397, Exercise 10.6: From “Pricing of Tuna,” Copyright 2001 by Consumers Union of U.S., Inc., Yonkers, NY 10703-1057, a nonproﬁt organization. Reprinted with permission from the June 2001 issue of Consumer Reports® for educational purposes only. No commercial use or reproduction permitted. www.ConsumerReports.org®; 446: From “Four-Day Work Week Improves Environment” by C.S. Catlin in Environmental Health, Vol. 59, No. 7, March 1997. Copyright 1997 National Environmental Health Association. Reprinted by permission. Chapter 11. 447: © James Leynse/CORBIS; 462, Exercise 11.16: From “Pricing of Tuna,” Copyright 2001 by Consumers Union of U.S., Inc., Yonkers, NY 10703-1057, a nonproﬁt organization. Reprinted with permission from the June 2001 issue of Consumer Reports® for educational purposes only. No commercial use or reproduction permitted. www.ConsumerReports.org®. Chapter 12. 502: © Justin Sullivan/Getty Images; 549, Exercise 12.80: From “Ratings: Walking Shoes,” Copyright 2006 by Consumers Union of U.S., Inc., Yonkers, NY 10703-1057, a nonproﬁt organization. Reprinted with permission from the October 2006 issue of Consumer
Reports® for educational purposes only. No commercial use or reproduction permitted. www.ConsumerReports.org®. Chapter 13. 551: © Will & Deni McIntyre/CORBIS; 590, Exercise 13.33: From “Tuna Goes Upscale,” Copyright 2001 by Consumers Union of U.S., Inc., Yonkers, NY 10703-1057, a nonproﬁt organization. Reprinted with permission from the June 2001 issue of Consumer Reports® for educational purposes only. No commercial use or reproduction permitted. www.ConsumerReports.org®. Chapter 14. 594: © Dave Bartruff/CORBIS; 601, Exercises 14.13, 14.14: M&M’s® and M® are registered trademarks owned by Mars, Incorporated and its affiliates. These trademarks are used with permission. Mars, Incorporated is not associated with Cengage Learning Market Group worldwide. © Mars, Inc. 2008. Chapter 15. 629: © Don Carstens/Brand X/CORBIS; 677: From “Eggs Substitutes Range in Quality” by K. Sakekel in The San Francisco Chronicle, Febuary 10, 1993, p. 8. Copyright © 1993 San Francisco Chronicle. Appendix. 691: From “Table of Percentage Points of the t-Distribution,” Biometrika 32 (1941):300. Reproduced by permission of the Biometrika Trustees; 692: From “Tables of the Percentage Points of the x2-Distribution,” Biometrika Tables for Statisticians, Vol. 1, 3rd ed. (1966). Reproduced by permission of the Biometrika Trustees; 694: A portion of “Tables of Percentage Points of the Inverted Beta (F) Distribution,” Biometrika, Vol. 33 (1943) by M. Merrington and C.M. Thompson and from Table 18 of Biometrika Tables for Statisticians, Vol. 1, Cambridge University Press, 1954, edited by E.S. Pearson and 746 ❍ CREDITS H.O. Hartley. Reproduced with permission of the authors, editors, and Biometrika Trustees; 702, Tables 7(a) and 7(b): Data from “An Extended Table of Critical Values for the Mann-Whitney (Wilcoxon) Two
-Sample Statistic” by Roy C. Milton, pp. 925–934, in the Journal of the American Statistical Association, Vol. 59, No. 307, Sept. 1964. Reprinted with permission from the Journal of the American Statistical Association. Copyright 1964 by the American Statistical Association. All rights reserved; 704: From “Some Rapid Approximate Statistical Procedures” (1964) 28, by F. Wilcoxon and R.A. Wilcox. Reproduced with the kind permission of Lederle Laboratories, a division of American Cyanamid Company; 705: From “Distribution of Sums of Squares of Rank Differences for Small Samples” by E.G. Olds, Annals of Mathematical Statistics 9 (1938). Reproduced with the permission of the editor, Annals of Mathematical Statistics; 706: From Handbook of Tables for Probability and Statistics, 2nd ed., edited by William H. Beyer (CRC Press). Used by permission of William H. Beyer. Answers to MyPersonal Trainer Exercises Chapter 1 A. 90 5.9 200 12.86.98 25 15 1.0 25 Chapter 2 A. B. 0 0 500 0 to < 15 15 to < 30 0 to < 1.0 1.0 to < 2.0 500 to < 525 525 to < 550 Data Set Sorted n Position of Q1 Position of Q3 Lower Quartile, Q1 Upper Quartile, Q3 2, 5, 7, 1, 1, 2, 8 1, 1, 2, 2, 5, 7, 8 7 5, 0, 1, 3, 1, 5, 5, 2, 4, 4, 1 0, 1, 1, 1, 2, 3, 4, 4, 5, 5, 5 11 2nd 3rd 6th 9th 1 1 7 5 B. Sorted Data Set Position of Q1 Adjacent Values 0, 1, 4, 4, 5, 9 0, 1, 3, 3, 4, 7, 7, 8 1.75 2.25 1, 1, 2, 5, 6, 6, 7, 9, 9 2.5 0 and 1 1 and 3 1 and 2 Q1 0.75(1).75 1.25(2) 1.5 1.5(1) 1.5 Position of Q3 Adjacent Values 5.25 6.75 7
.5 5 and 9 7 and 7 7 and 9 Q3 5.25(4) 6 7.75(0) 7 7.5(2) 8 Chapter 3 A. x 0 2 4 y 1 5 2 xy 0 10 8 Sx 6 Sy 8 Sxy 18 Calculate: n 3 sx 2 sy 2.082 Covariance (Sy) Sxy (Sx) n n 1 1 sxy Correlation Coefficient s r.240 x y s s x y B. From Part A Sx 6 From Part A sx 2 Calculate: Slope x 2 y-Intercept a y bx 2.167.25 b rs y s x Sy 8 sy 2.082 r.240 y 2.667 Regression Line: y 2.167.25x Chapter 4 P(A) P(B) Conditions for Events A and B P(A B) P(A B) P(AB).3.3.1.2.4.4.5.5 Mutually exclusive Independent Mutually exclusive and dependent Independent 0.12 0.10.7.58.6.6 0.3 0.2 Chapter 5 Section 5.2 A. B..010,.087,.317,.663,.922, 1.000 0, 1, 2, 3, 4 4, 5 5 0, 1, 2, 3 2, 3, 4 4 P(x 4) P(x 4) P(x 4) P(x 4) P(2 x 4) P(x 4) n/a 1 P(x 3) 1 P(x 4) P(x 3) P(x 4) P(x 1) P(x 4) P(x 3).922.337.078.663.835.259 Chapter 5 Section 5.3 A. B. C..223,.558,.809,.934,.981,.996,.999, 1.000 1.5 1.50 e,.223! 0 1.5 1.51 e,.335 1! 0, 1,.558 0, 1, 2, 3 3, 4, 5,... 4, 5, 6,... 0, 1, 2 2, 3, 4 3 P(x 3) P(x 3) P(x 3) P(x 3) P
(2 x 4) P(x 3) n/a 1 P(x 2) 1 P(x 3) P(x 2) P(x 4) P(x 1) P(x 3) P(x 2).934.191.066.809.423.125 Chapter 6 Section 6.3 1.5 2 2.33 1.96, 1.96 1.24, 2.37 1 n/a 1 P(z 2) 1 P(z 2.33) P(z 1.96) P(z 1.96) P(z 2.37) P(z 1.24) n/a.9332 1.9772.0228 1.9901.0099.9750.0250.9500.9911.1075.8836.1587 Chapter 6 Section 6.4 A. B. 1. 12; 18 2. yes 3. 12; 2.683 1. 20, 21,..., 30 2. 20; 19.5 3. 2.80 4. 2.80;.9974;.0026 Chapter 7 Section 7.5 A. B. C. normal; 75; 2 P(x 80); 2.5; 80; 2.5;.9938;.0062 P(70 x 72); 2.5; 1.5; 70; 72; 2.5; 1.5;.0668;.0062;.0606 Chapter 7 Section 7.6 A. B. C. normal;.4;.08165 P(pˆ.5); 1.22;.5; 1.22;.8888;.1112 P(.5 pˆ.6); 1.22; 2.45;.5;.6; 1.22; 2.45;.9929;.8888;.1041 Chapter 8 Type 1 or 2 MOE q 1.96p n Quantitative One Solve.6).1 1.96.4( n 6 1 1.96 n 2 1.963 6 3n Binomial Two.6).05.6).4( 1.96.4( n n Sample Size n 93 n 139 n1 70 n2 70 n1 738 n2 738 Chapter 9 A. B. Critical Value Rejection Region Conclusion p-value p-value a?
Conclusion 1.645 2.33 1.96 2.58 z 1.645 z 2.33 z 1.96 or z 1.96 Do not reject H0 z 2.58 or z 2.58 Reject H0 Do not reject H0 Reject H0.0808.0069.4592 0 No Yes No Yes Do not reject H0 Reject H0 Do not reject H0 Reject H0i = 0. A curve is irreducible if its equation is an irreducible polynomial. The decomposition X = X1 ∪ · · · ∪ Xr just obtained is called a decomposition of X into irreducible components. · · · f kr r In certain cases, the notions just introduced turn out not to be well deﬁned, or to differ wildly from our intuition. This is due to the speciﬁc nature of the ﬁeld k in I.R. Shafarevich, Basic Algebraic Geometry 1, DOI 10.1007/978-3-642-37956-7_1, © Springer-Verlag Berlin Heidelberg 2013 3 4 1 Basic Notions which the coordinates of points of the curve are taken. For example if k = R then following the above terminology we should call the point (0, 0) a “curve”, since it is deﬁned by the equation x2 + y2 = 0. Moreover, this “curve” should have “degree” 2, but also any other even number, since the same point (0, 0) is also deﬁned by the equation x2n + y2n = 0. The curve is irreducible if we take its equation to be x2 + y2 = 0, but reducible if we take it to be x6 + y6 = 0. Problems of this kind do not arise if k is an algebraically closed ﬁeld. This is based on the following simple fact. Lemma Let k be an arbitrary ﬁeld, f ∈ k[x, y] an irreducible polynomial, and g ∈ k[x, y] an arbitrary polynomial. If g is not divisible by f then the system of equations f (x, y) = g(x, y) = 0 has only a
ﬁnite number of solutions. Proof Suppose that x appears in f with positive degree. We view f and g as elements of k(y)[x], that is, as polynomials in one variable x, whose coefﬁcients are rational functions of y. It is easy to check that f remains irreducible in this ring: if f splits as a product of factors, then after multiplying each factor by the common denominator a(y) ∈ k[y] of its coefﬁcients, we obtain a relation that contradicts the irreducibility of f in k[x, y]. For the same reason, g is not divisible by f in the new ring k(y)[x]. Hence there exist two polynomials u,v ∈ k(y)[x] such that f u + gv = 1. Multiplying this equality through by the common denominator a ∈ k[y] of all the coefﬁcients of u and v gives f u + gv = a, where u = au, v = av ∈ k[x, y], and 0 = a ∈ k[y]. It follows that if f (α, β) = g(α, β) = 0 then a(β) = 0, that is, there are only ﬁnitely many possible values for the second coordinate β. For each such value, the ﬁrst coordinate α is a root of f (x, β) = 0. The polynomial f (x, β) is not identically 0, since otherwise f (x, y) would be divisible by y − β, and hence there are also only a ﬁnite number of possibilities for α. The lemma is proved. An algebraically closed ﬁeld k is inﬁnite; and if f is not a constant, the curve with equation f (x, y) = 0 has inﬁnitely many points. Because of this, it follows from the lemma that an irreducible polynomial f (x, y) is uniquely determined, up to a constant multiple, by the curve f (x, y) = 0. The same holds for an arbitrary polynomial, under the assumption that its factorisation into irreducible components has no multiple factors. We can always choose the equation of
a curve to be a polynomial satisfying this condition. The notion of the degree of a curve, and of an irreducible curve, is then well deﬁned. Another reason why algebraic geometry only makes sense on passing to an algebraically closed ﬁeld arises when we consider the number of points of intersection of curves. This phenomenon is already familiar from algebra: the theorem that the number of roots of a polynomial equals its degree is only valid if we consider roots in an algebraically closed ﬁeld. A generalisation of this theorem is the so-called Bézout theorem: the number of points of intersection of two distinct irreducible algebraic curves equals the product of their degrees. The lemma shows that, in any 1 Algebraic Curves in the Plane 5 Figure 1 Intersections of conics case, this number is ﬁnite. The theorem on the number of roots of a polynomial is a particular case, for the curves y − f (x) = 0 and y = 0. Bézout’s theorem holds only after certain amendments. The ﬁrst of these is the requirement that we consider points with coordinates in an algebraically closed ﬁeld. Thus Figure 1 shows three cases for the relative position of two curves of degree 2 (ellipses) in the real plane. Here Bézout’s theorem holds in case (c), but not in cases (a) and (b). We assume throughout what follows that k is algebraically closed; in the contrary case, we always say so. This does not mean that algebraic geometry does not apply to studying questions concerned with algebraically nonclosed ﬁelds k0. However, applications of this kind most frequently involve passing to an algebraically closed ﬁeld k containing k0. In the case of R, we pass to the complex number ﬁeld C. This often allows us to guess or to prove purely real relations. Here is the most elementary example of this nature. If P is a point outside a circle C then there are two tangent lines to C through P. The line joining their points of contact is called the polar line of P with respect to C (Figure 2, (a)). All these constructions can be expressed in terms of algebraic relations between the coordinates of P and the equation of C. Hence they are also applicable to
the case that P lies inside C. Of course, the points of tangency of the lines now have complex coordinates, and can’t be seen in the picture. But since the original data was real, the set of points obtained (that is, the two points of tangency) should be invariant on replacing all the numbers by their complex conjugates; that is, the two points of tangency are complex conjugates. Hence the line L joining them is real. This line is also called the polar line of P with respect to C. It is also easy to give a purely real deﬁnition of it: it is the locus of points outside the circle whose polar line passes through P (Figure 2, (b)). Here are some other situations in which questions arise involving algebraic geometry over an algebraically nonclosed ﬁeld, and whose study usually requires passing to an algebraically closed ﬁeld. (1) k = Q. The study of points of an algebraic curve f (x, y) = 0, where f ∈ Q[x, y], and the coordinates of the points are in Q. This is one of the fundamental problems of number theory, the theory of indeterminate equations. For example, Fermat’s last theorem requires us to describe points (x, y) ∈ Q2 of the curve xn + yn = 1. 6 1 Basic Notions Figure 2 The polar line of a point with respect to a conic (2) Finite ﬁelds. Let k = Fp be the ﬁeld of residues modulo p. Studying the points with coordinates in k on the algebraic curve given by f (x, y) = 0 is another problem of number theory, on the solutions of the congruence f (x, y) ≡ 0 mod p. (3) k = C(z). Consider the algebraic surface in A3 given by F (x, y, z) = 0, with F (x, y, z) ∈ C[x, y, z]. By putting z into the coefﬁcients and thinking of F as a polynomial in x, y, we can consider our surface as a curve over the ﬁeld C(z) of rational functions in z. This is an extremely fertile method in the study of algebraic surfaces. 1.2 Rational
Curves As is well known, the curve given by y2 = x2 + x3 (1.2) has the property that the coordinates of its points can be expressed as rational functions of one parameter. To deduce these expressions, note that the line through the origin y = tx intersects the curve (1.2) outside the origin in a single point. Indeed, substituting y = tx in (1.2), we get x2(t 2 − x − 1) = 0; the double root x = 0 corresponds to the origin 0 = (0, 0). In addition to this, we have another root x = t 2 − 1; the equation of the line gives y = t (t 2 − 1). We thus get the required parametrisation x = t 2 − 11.3) and its geometric meaning is evident: t is the slope of the line through 0 and (x, y); and (x, y) are the coordinates of the point of intersection of the line y = tx with the curve (1.2) outside 0. We can see this parametrisation even more intuitively by drawing another line, not passing through 0 (for example, the line x = 1) and projecting the curve from 0, by sending a point P of the curve to the point Q of intersection of the line 0P with this line (see Figure 3). Here the parameter t plays the role of coordinate on the given line. Either from this geometric description, or from (1.3), we see that t is uniquely determined by the point (x, y) (for x = 0). 1 Algebraic Curves in the Plane 7 Figure 3 Projection of a cubic We now give a general deﬁnition of algebraic plane curves for which a representation in these terms is possible. We say that an irreducible algebraic curve X deﬁned by f (x, y) = 0 is rational if there exist two rational functions ϕ(t) and ψ(t), at least one nonconstant, such that ϕ(t), ψ(t) f ≡ 0, (1.4) as an identity in t. Obviously if t = t0 is a value of the parameter, and is not one of the ﬁnitely many values at which the denominator of ϕ or ψ vanishes, then (ϕ(t0), ψ(t0
)) is a point of X. We will show subsequently that for a suitable choice of the parametrisation ϕ, ψ, the map t0 → (ϕ(t0), ψ(t0)) is a one-to-one correspondence between the values of t and the points of the curve, provided that we exclude certain ﬁnite sets from both the set of values of t and the points of the curve. Then conversely, the parameter t can be expressed as a rational function t = χ(x, y) of the coordinates x and y. If the coefﬁcients of the rational functions ϕ and ψ belong to some subﬁeld k0 of k and t0 ∈ k0 then the coordinates of the point (ϕ(t0), ψ(t0)) also belong to k0. This observation points to one possible application of the notion of rational curve. Suppose that f (x, y) has rational coefﬁcients. If we know that the curve given by (1.1) is rational, and that the coefﬁcients of ϕ and ψ are in Q, then the parametrisation x = ϕ(t), y = ψ(t) gives us all the rational points of this curve, except possibly a ﬁnite number, as t runs through all rational values. For example, all the rational solutions of the indeterminate equation (1.2) can be obtained from (1.3) as t runs through all rational values. Another application of rational curves relates to integral calculus. We can view the equation of the curve (1.1) as determining y as an algebraic function of x. Then any rational function g(x, y) is a (usually complicated) function of x. The rationality of the curve (1.1) implies the following important fact: for any rational function g(x, y), the indeﬁnite integral 8 Figure 4 Projection of a conic 1 Basic Notions g(x, y)dx (1.5) can be expressed in elementary functions. Indeed, since the curve is rational, it can be parametrised as x = ϕ(t), y = ψ(t) where ϕ, ψ are rational functions. Substituting these expressions in the integral (1.5), we reduce it to the
form g(ϕ(t), ψ(t))ϕ(t)dt, which is an integral of a rational function. It is known that an integral of this form can be expressed in elementary functions. Substituting the expression t = χ(x, y) for the parameter in terms of the coordinates, we get an expression for the integral (1.5) as an elementary function of the coordinates. We now give some examples of rational curves. Curves of degree 1, that is, lines, are obviously rational. Let us prove that an irreducible conic X is rational. Choose a point (x0, y0) on X. Consider the line through (x0, y0) with slope t. Its equation is y − y0 = t (x − x0). (1.6) We ﬁnd the points of intersection of X with this line; to do this, solve (1.6) for y and substitute this in the equation of X. We get the equation for x x, y0 + t (x − x0) f = 0, (1.7) which has degree 2, as one sees easily. We know one root of this quadratic equation, namely x = x0, since by assumption (x0, y0) is on the curve. Divide (1.7) by the coefﬁcient of x2, and write A for the coefﬁcient of x in the resulting equation; the other root is then determined by x + x0 = −A. Since t appears in the coefﬁcients of (1.7), A is a rational function of t. Substituting the expression x = −x0 − A in (1.6), we get an expression for y also as a rational function of t. These expressions for x and y satisfy the equation of the curve, as can be seen from their derivation, and thus prove that the curve is rational. The above parametrisation has an obvious geometric interpretation. A point (x, y) of X is sent to the slope of the line joining it to (x0, y0); and the parameter t is sent to the point of intersection of the curve with the line through (x0, y0) with slope t. This point is uniquely determined precisely because we are dealing with an irreducible curve of degree 2. In the same
way as the parametrisation of the curve (1.2), this parametrisation can be interpreted as the projection of X from the point (x0, y0) to some line not passing through this point (Figure 4). Note that in constructing the parametrisation we have used a point (x0, y0) of X. If the coefﬁcients of the polynomial f (x, y) and the coordinates of (x0, y0) are 1 Algebraic Curves in the Plane 9 contained in some subﬁeld k0 of k, then so do the coefﬁcients of the functions giving the parametrisation. Thus we can, for example, ﬁnd the general form for the solution in rational numbers of an indeterminate equation of degree 2 if we know just one solution. The question of whether there exists one solution is rather delicate. For the rational number ﬁeld Q it is solved by Legendre’s theorem (see for example Borevich and Shafarevich [15, Section 7.2, Chapter 1]). √ We consider another application of the parametrisation we have found. The second degree equation y2 = ax2 + bx + c deﬁnes a rational curve, as we have just seen. It follows from this that for any rational function g(x, y), the integral ax2 + bx + c)dx can be expressed in elementary functions. The parametrisation we have given provides an explicit form of the substitutions that reduce this integral to an integral of a rational function. It is easy to see that this leads to the well-known Euler substitutions. g(x, The examples considered above lead us to the following general question: how can we determine whether an arbitrary algebraic plane curve is rational? This question relates to quite delicate ideas of algebraic geometry, as we will see later. 1.3 Relation with Field Theory We now show how the question at the end of Section 1.2 can be formulated as a problem of ﬁeld theory. To do this, we assign to every irreducible plane curve a certain ﬁeld, by analogy with the way we assign to an irreducible polynomial in one variable the smallest ﬁeld extension in which it has a root. Let X be the irreducible
curve given by (1.1). Consider rational functions u(x, y) = p(x, y)/q(x, y), where p and q are polynomials with coefﬁcients in k such that the denominator q(x, y) is not divisible by f (x, y). We say that such a function u(x, y) is a rational function deﬁned on X; and two rational functions p(x, y)/q(x, y) and p1(x, y)/q1(x, y) deﬁned on X are equal on X if the polynomial p(x, y)q1(x, y) − q(x, y)p1(x, y) is divisible by f (x, y). It is easy to check that rational functions on X, up to equality on X, form a ﬁeld. This ﬁeld is called the function ﬁeld or ﬁeld of rational functions of X, and denoted by k(X). A rational function u(x, y) = p(x, y)/q(x, y) is deﬁned at all points of X where q(x, y) = 0. Since by assumption q is not divisible by f, by Lemma of Section 1.1, there are only ﬁnitely many points of X at which u(x, y) is not deﬁned. Hence we can also consider elements of k(X) as functions on X, but deﬁned everywhere except at a ﬁnite set. It can happen that a rational function u has two different expressions u = p/q and u = p1/q1, and that for some point (α, β) ∈ X we have q(α, β) = 0 but q1(α, β) = 0. For example, the function u = (1 − y)/x on the circle x2 + y2 = 1 at the point (0, 1) has an alternative expression u = x/(1 + y) whose denominator does not vanish at (0, 1). If u has an expression u = p/q with q(P ) = 0 then we say that u is regular at P. 10 1 Basic Notions Every element
of k(X) can obviously be written as a rational function of x and y; now x, y are algebraically dependent, since they are related by f (x, y) = 0. It is easy to check from this that k(X) has transcendence degree 1 over k. If X is a line, given say by y = 0, then every rational function ϕ(x, y) on X is a rational function ϕ(x, 0) of x only, and hence the function ﬁeld of X equals the ﬁeld of rational functions in one variable, k(X) = k(x). Now assume that the curve X is rational, say parametrised by x = ϕ(t), y = ψ(t). Consider the substitution u(x, y) → u(ϕ(t), ψ(t)) that takes any rational function u = p(x, y)/q(x, y) on X into the rational function in t obtained by substituting ϕ(t) for x and ψ(t) for y. We check ﬁrst that this substitution makes sense, that is, that the denominator q(ϕ(t), ψ(t)) is not identically 0 as a function of t. Assume that q(ϕ(t), ψ(t)) = 0, and compare this equality with (1.4). Recalling that the ﬁeld k is algebraically closed, and therefore inﬁnite, by making t take different values in k, we see that f (x, y) = 0 and q(x, y) = 0 have inﬁnitely many common solutions. But by Lemma of Section 1.1, this is only possible if f and q have a common factor. Thus our substitution sends any rational function u(x, y) deﬁned on X into a well-deﬁned element of k(t). Moreover, since ϕ and ψ satisfy the relation (1.4), the substitution takes rational functions u, u1 that are equal on X to the same rational function in t. Thus every element of k(X) goes to a well-deﬁned element of k(t). This map is obviously an isomorphism of k(X) with some subﬁeld of k
(t). It takes an element of k to itself. At this point we make use of a theorem on rational functions. This is the result known as Lüroth’s theorem, that asserts that a subﬁeld of the ﬁeld k(t) of rational functions containing k is of the form k(g(t)), where g(t) is some rational function; that is, the subﬁeld consists of all the rational functions of g(t). If g(t) is not constant, then sending f (u) → f (g(t)) obviously gives an isomorphism of the ﬁeld of rational functions k(u) with k(g(t)). Thus Lüroth’s theorem can be given the following statement: a subﬁeld of the ﬁeld of rational functions k(t) that contains k and is not equal to k is itself isomorphic to the ﬁeld of rational functions. Lüroth’s theorem can be proved from simple properties of ﬁeld extensions (see van der Waerden [76, 10.2 (Section 73)]). Applying it to our situation, we see that if X is a rational curve then k(X) is isomorphic to the ﬁeld of rational functions k(t). Suppose, conversely, that for some curve X given by (1.1), the ﬁeld k(X) is isomorphic to the ﬁeld of rational functions k(t). Suppose that under this isomorphism x corresponds to ϕ(t) and y to ψ(t). The polynomial relation f (x, y) = 0 ∈ k(X) is respected by the ﬁeld isomorphism, and gives f (ϕ(t), ψ(t)) = 0; therefore X is rational. It is easy to see that any ﬁeld K ⊃ k having transcendence degree 1 over k and generated by two elements x and y is isomorphic to a ﬁeld k(X), where X is some irreducible algebraic plane curve. Indeed, x and y must be connected by a polynomial relation, since K has transcendence degree 1 over k. If this dependence relation is f (x, y) = 0, with
f an irreducible polynomial, then we can obviously take X to be the algebraic curve deﬁned by this equation. It follows from this that the question on rational curves posed at the end of Section 1.2 is equivalent to the following question of ﬁeld theory: when is a ﬁeld K ⊃ k with transcendence degree 1 over k and generated by two elements x and y isomorphic to the ﬁeld of rational functions 1 Algebraic Curves in the Plane 11 of one variable k(t)? The requirement that K is generated over k by two elements is not very natural from the algebraic point of view. It would be more natural to consider ﬁeld extensions generated by an arbitrary ﬁnite number of elements. However, we will prove later that doing this does not give a more general notion (compare Theorem 1.8 and Proposition A.7). In conclusion, we note that the preceding arguments allow us to solve the problem of obtaining a generically one-to-one parametrisation of a rational curve. Let X be a rational curve. By Lüroth’s theorem, the ﬁeld k(X) is isomorphic to the ﬁeld of rational functions k(t). Suppose that this isomorphism takes x to ϕ(t) and y to ψ(t). This gives the parametrisation x = ϕ(t), y = ψ(t) of X. Proposition The parametrisation x = ϕ(t), y = ψ(t) has the following properties: (i) Except possibly for a ﬁnite number of points, any (x0, y0) ∈ X has a represen- tation (x0, y0) = (ϕ(t0), ψ(t0)) for some t0. (ii) Except possibly for a ﬁnite number of points, this representation is unique. Proof Suppose that the function that maps to t under the isomorphism k(X) → k(t) is χ(x, y). Then the inverse isomorphism k(t) → k(X) is given by the formula u(t) → u(χ(x, y)). Writing out the fact that the correspondences are inverse to one another gives �
�(x, y(t), ψ(t) t = χ χ(x, y), (1.8) (1.9) Now (1.8) implies (i). Indeed, if χ(x, y) = p(x, y)/q(x, y) and q(x0, y0) = 0, we can take t0 = χ(x0, y0); there are only ﬁnitely many points (x0, y0) ∈ X at which q(x0, y0) = 0, since q(x, y) and f (x, y) are coprime. Suppose that (x0, y0) is such that χ(x0, y0) is distinct from the roots of the denominators of ϕ(t) and ψ(t); there are only ﬁnitely many points (x0, y0) for which this fails, for similar reasons. Then formula (1.8) gives the required representation of (x0, y0). In the same way, it follows from (1.9) that the value of the parameter t, if it exists, is uniquely determined by the point (x0, y0), except possibly for the ﬁnite number of points at which q(x0, y0) = 0. The proposition is proved. Note that we have proved (i) and (ii) not for any parametrisation of a rational curve, but for a specially constructed one. For an arbitrary parametrisation, (ii) can be false: for example, the curve (1.2) has, in addition to the parametrisation given by (1.3), another parametrisation x = t 4 − 1, y = t 2(t 4 − 1), obtained from (1.3) on replacing t by t 2. Obviously here the values t and −t of the parameter correspond to the same point of the curve. 1.4 Rational Maps A rational parametrisation is a particular case of a more general notion. Let X and Y be two irreducible algebraic plane curves, and u, v ∈ k(X). The map ϕ(P ) = 12 1 Basic Notions (u(P ), v(P )) is deﬁned at all points P of X
where both u and v are deﬁned; it is called a rational map from X to Y if ϕ(P ) ∈ Y for every P ∈ X at which ϕ is deﬁned. If Y has the equation g = 0 then g(u, v) ∈ k(X) must vanish at all but ﬁnitely many points of X, and therefore we must have g(u, v) = 0 ∈ k(X). For example, the projection from a point P considered in Section 1.2 is a rational map of X to the line. A rational parametrisation of a rational curve X is a rational map of the line to X. A rational map ϕ : X → Y is birational, or is a birational equivalence of X to Y, if ϕ has a rational inverse, that is, if there exists a rational map ψ : Y → X such that ϕ ◦ ψ and ψ ◦ ϕ are the identity (at the points where they are deﬁned). In this case, we say that X and Y are birational, or birationally equivalent. A birational map is not constant, that is, at least one of the functions deﬁning it is not an element of k. Indeed, a constant map is deﬁned everywhere, and sends X to a single point Q ∈ Y. Taking any point Q = Q at which the inverse ψ of ϕ is deﬁned contradicts the deﬁnition. It follows that for any point Q ∈ Y the inverse image ϕ−1(Q) of Q (the set of points P ∈ X such that ϕ(P ) = Q) is ﬁnite; this follows at once from Lemma of Section 1.1. Let S be the ﬁnite set of points of X at which a birational map ϕ : X → Y is not deﬁned, U = X \ S its complement, and T and V the same for ψ : Y → X. It follows from what we said above that the complement in X of ϕ−1(V ) ∩ U and in Y of ψ −1(U ) ∩ V are ﬁnite, and ϕ establishes a one-to-one correspondence between ϕ−1(V
) ∩ U and ψ −1(U ) ∩ V. Birational equivalence is a fundamental equivalence relation in algebraic geometry, and we usually classify algebraic curves up to birational equivalence. We have seen that the rational curves are exactly the curves birational to the line. Suppose that the equation f (x, y) of an irreducible curve of degree n is a polynomial all of whose terms are monomials in x and y of degree n − 1 and n only. Then the projection from the origin deﬁnes a birational map of our curve and the line: this can be proved by a direct generalisation of the arguments for the curve (1.2). Now suppose that the equation f has terms of degrees n − 2, n − 1 and n, that is, f = un−2 + un−1 + un, where ui is homogeneous of degree i. Again we set y = tx and cancel the factor of xn−2 from the equation, thus reducing it to the form a(t)x2 + b(t)x + c(t) = 0, where a(t) = un(1, t), b(t) = un−1(1, t) and c(t) = un−2(1, t). Setting s = 2ax + b to complete the square (assuming that the ground ﬁeld has characteristic = 2), we see that our curve is birational to the curve given by s2 = p(t), where p = b2 − 4ac. A curve of this type is called a hyperelliptic curve. If p(t) has even degree 2m then rewriting it in the form p(t) = q(t)(t − α) and dividing both sides of the equation through by (t − α)2m shows that the curve is birational to the curve given by η2 = h(ξ ), where t − α)m and h(ξ ) = q(t) (t − α)2m−1, in which h is a polynomial of degree ≤2m − 1 in ξ. 1 Algebraic Curves in the Plane 13 These ideas apply in particular to any cubic curve, if we take the origin to be any point of the curve. We see that, if char k = 2, an irreducible cubic curve is
birational to a curve given by y2 = f (x) where f is a polynomial of degree ≤3. If f (x) has degree ≤2 then the cubic is rational. If it has degree 3 then we can assume that its leading coefﬁcient is 1. Then the equation takes the form y2 = x3 + ax2 + bx + c. This is called the Weierstrass normal form of the equation of a cubic. If char k = 3 then after making a translation x → x − a/3 we can reduce the equation to the form y2 = x3 + px + q. (1.10) Let X and Y be two irreducible algebraic plane curves that are birational, and suppose that the maps between them are given by (u, v) = ϕ(x, y), ψ(x, y) and (x, y) = ξ(u, v), η(u, v). As in our study of rational curves, we can establish a relation between the function ﬁelds k(X) and k(Y ) of these two curves. For this, we send a rational function w(x, y) ∈ k(X) to w(ξ(u, v), η(u, v)), viewed as a rational function on Y. It is easy to check that this deﬁnes a map k(X) → k(Y ) that is an isomorphism between these two ﬁelds. Conversely, if k(X) and k(Y ) are isomorphic, then under this isomorphism x, y ∈ k(X) correspond to functions ξ(u, v), η(u, v) ∈ k(Y ), and u, v ∈ k(Y ) to functions ϕ(x, y), ψ(x, y) ∈ k(X), and it is again trivial to check that the pairs of functions ϕ, ψ and ξ, η deﬁne birational maps between the curves X and Y. Thus two curves are birational if and only if their rational function ﬁelds are isomorphic. We see that the problem of classifying algebraic curves up to birational equivalence is a geometric aspect of the natural algebraic problem of classifying ﬁ
nitely generated extension ﬁelds of k of transcendence degree 1 up to isomorphism. In this problem, it is also natural not to restrict to ﬁelds of transcendence degree 1, but to consider ﬁelds of any ﬁnite transcendence degree. We will see later that this wider formulation of the problem also has a geometric interpretation. However, for this we have to leave the framework of the theory of algebraic curves, and consider algebraic varieties of any dimension. 1.5 Singular and Nonsingular Points We borrow a deﬁnition from coordinate geometry: a point P is a singular point or singularity of the curve deﬁned by f (x, y) = 0 if f y(P ) = f (P ) = 0, where f x denotes the partial derivative ∂f/∂x. If we translate P to the origin, we can say that (0, 0) is singular if f does not have constant or linear terms. A point x(P ) = f 14 Figure 5 A cusp 1 Basic Notions is nonsingular if it is not singular, that is, if f y(P ) = 0. A curve all of whose points are nonsingular is nonsingular or smooth. It is well known that an irreducible conic is nonsingular; the simplest example of a singular curve is the curve of (1.2). x(P ) or f For an irreducible curve, either f x is divisible by f. However, since f x vanishes at only ﬁnitely many points of the curve, or f x has smaller degree than f, the = 0. The same holds for f latter is only possible if f = f = 0 implies, x y if char k = 0, that f ∈ k, and, if char k = p > 0, that f involves x and y only as pth powers; in this last case, taking pth roots of the coefﬁcients of f and using the well-known characteristic p identity (α + β)p = αp + βp, we deduce that y. But f x p f = aij xpiypj = bij xiyj where bp ij = aij, which contradicts the irreducibility of the curve. This shows that an irreduc
ible curve has only a ﬁnite number of singular points. If P = (0, 0) and the leading terms in the equation of the curve have degree r, then r is called the multiplicity of P, and we say that P is an r-tuple point, or point of multiplicity r. Thus a nonsingular point has multiplicity 1. If P = (0, 0) has multiplicity 2 and the terms of degree 2 in the equation of the curve are ax2 + bxy + cy2 then there are two possibilities: (a) ax2 + bxy + cy2 factorises into two distinct linear factors; or (b) ax2 + bxy + cy2 is a perfect square. In case (a) the singularity is called a node (see Figure 3), and in case (b) a cusp (Figure 5). It follows from the deﬁnition that a curve of degree n cannot have a singularity of multiplicity >n. If a singular point has multiplicity n then the equation of the curve is a homogeneous polynomial in x and y of degree n, and therefore factorises as a product of linear factors, so that the curve is reducible. In Section 1.4 we proved that if an irreducible curve of degree n has a point of multiplicity n − 1 it is rational, and if it has a point of multiplicity n − 2 then it is hyperelliptic. The cubic curve written in Weierstrass normal form (1.10) is nonsingular if and only if the cubic polynomial on the right-hand side has no multiple roots, that is, 4p3 + 27q2 = 0. In this case it is called an elliptic curve. If k = R and P is a nonsingular point of the curve with equation f (x, y) = 0, and f y(P ) = 0, say, then by the implicit function theorem we can write y as a function of x in some neighbourhood of P. Substituting this expression for y, this represents any rational function on the curve as a function of x near P. When k is a general ﬁeld, x can still be used to describe all the rational functions on the curve, admittedly to a more modest extent. For simplicity, set P = (0, 0). Then f = αx + βy + g, where
g contains only terms of degree ≥2 and β = 0. We distinguish the terms in f that involve x only, writing f = xϕ(x) + yβ + yh, 1 Algebraic Curves in the Plane 15 with h(0, 0) = 0. Thus on the curve f = 0 we have y(β + h) = −xϕ(x), or, in other words, y = xv, where v = −ϕ(x)/(β + h) is a regular function at P (because β + h(P ) = 0). Let u be any rational function on our curve that is regular at P and has u(P ) = 0. Then u = p/q, where p, q ∈ k[x, y] with p(P ) = 0 and q(P ) = 0. Substituting our expression for y in this gives p(x, y) = p(x, xv) = xr (because p has no constant term), where r is a regular function on the curve, and hence u = xr/q = xu1. If u1(P ) = 0 then we can repeat the argument, getting u = x2u2, and so on. We now prove that, provided u is not identically 0 on the curve, this process must stop after a ﬁnite number of steps. For this, return to the expression u = p/q, in which, by assumption, p is not divisible by f. Hence there exist ξ, η ∈ k[x, y] and a polynomial a ∈ k[x] with a = 0 such that f ξ + pη = a (we have already used this argument in the proof of Lemma of Section 1.1). Suppose a = xka0 with a0(0) = 0. Then pη = a on the curve, and a representation p = xlw with l > k would give a contradiction: xk(xl−kw − a0) = 0 on the curve, that is, xl−kw − a0 = 0. If w = c/d with c, d ∈ k[x, y] and d(P ) = 0 then xl−kc − a0d = 0 on the curve, that is, xl−kc − a0
d is divisible by f. But this is impossible, since xl−k vanishes at P and a0d does not. Since any rational function is a ratio of regular functions, we have proved the following theorem. Theorem 1.1 At any nonsingular point P of an irreducible algebraic curve, there exists a regular function t that vanishes at P and such that every rational function u that is not identically 0 on the curve can be written in the form u = t kv, (1.11) with v regular at P and v(P ) = 0. The function u is regular at P if and only if k ≥ 0 in (1.11). A function t with this property is called a local parameter on the curve at P. Obviously two different local parameters are related by t = tv, where v is regular at P and v(P ) = 0. We saw in the proof of the theorem that if f y(P ) = 0 then x can be taken as a local parameter. The number k in (1.11) is called the multiplicity of the zero of u at P. It is independent of the choice of the local parameter. Let X and Y be algebraic curves with equations f = 0 and g = 0, and suppose that X is irreducible and not contained in Y, and that P ∈ X ∩ Y is a nonsingular point of X. Then g deﬁnes a function on X that is not identically zero; the multiplicity of the zero of g at P is called the intersection multiplicity2 of X and Y at P. The notion of intersection multiplicity is one of the amendments needed in a correct 2This is discussed at length later in the book; see Section 1.1, Chapter 4 for the general deﬁnition of intersection multiplicity, which is symmetric in X and Y, and for the fact that it coincides with the simple notion used here. 16 1 Basic Notions statement of Bézout’s theorem: for the theorem that the number of roots of a polynomial is equal to its degree is false unless we count roots with their multiplicities. Here we analyse intersection multiplicities in the case that X is a line. Let P = (α, β) ∈ X, and suppose that the equation of X is written in the form f (x, y) = a(x − α)
+ b(y − β) + g, where the polynomial g expanded in powers of x − α and y − β has only terms of degree ≥2. We write the equation of a line L through P in the form x = α + λt, y = β + μt. (1.12) t is a local parameter on L at P. The restriction of f to L is of the form f (α + λt, β + μt) = (aλ + bμ)t + t 2ϕ(t). From this we see that if P is singular, that is, if a = b = 0, then every line through P has intersection multiplicity >1 with X at P. On the other hand, if the curve is nonsingular, then there is only one such line, namely that for which aλ + bμ = 0, with equation a(x − α) + b(y − β) = 0. Obviously a = f y(P ), and hence this equation can we expressed x(P ), b = f x(P )(x − α) + f f y(P )(y − β) = 0. (1.13) The line given by this equation is called the tangent line to X at the nonsingular point P. We now determine when a line has intersection multiplicity ≥3 with a curve at a nonsingular point P = (α, β). For this, we write the equation in the form f (x, y) = a(x − α) + b(y − β) + c(x − α)2 + d(x − α)(y − β) + e(y − β)2 + h, (1.14) where h is a polynomial which has only terms of degree ≥3 when expanded in power of x − α and y − β. Restricting f to the line L given by (1.12), we get that f = (aλ + bμ)t + (cλ2 + dλμ + eμ2)t 2 + t 3ψ(t). Therefore the intersection multiplicity will be ≥3 if the two conditions aλ + bμ = cλ2 + dλμ + eμ2 = 0 hold. The ﬁrst of these, as we have seen, means that L is the tangent line to X at P,
and the second that moreover cu2 + duv + ev2 is divisible by au + bv as a homogeneous polynomial in u, v. Together they show that q = au + bv + cu2 + duv + ev2 is reducible: it is divisible by au + bv. Conversely, if q is reducible, then q = rs, and r and s must have degree 1, and one of them, say r, must vanish when u = v = 0. But then r is proportional to au + bv and cu2 + duv + ev2 is divisible by it. Thus the reducibility of the conic q = au + bv + cu2 + duv + ev2 is a necessary and sufﬁcient condition for there to exist a line L through P with intersection multiplicity ≥3 at P. Such a point is called an inﬂexion point or ﬂex of X. We know from coordinate geometry the condition for a conic to be reducible. y(P ), We assume that k has characteristic = 2; then recalling that a = f x(P ), b = f 1 Algebraic Curves in the Plane 17 c = (1/2)f the form x (P ), d = f xy(P ) and e = (1/2)f yy(P ), we can write this condition in f xx f xy f x f xy f yy f y f x f y 0 (P ) = 0. (1.15) 1.6 The Projective Plane We return to Bézout’s theorem stated in Section 1.1. Even if we consider points with coordinates in an algebraically closed ﬁeld and take account of multiplicities of intersections, this fails in very simple cases, and still needs one further amendment. This can already be seen in the example of two lines, which have no points of intersection if they are parallel. However, on the projective plane, parallel lines do intersect, in a point of the line at inﬁnity. In the same way, any two circles in the plane, although they are curves of degree 2, have at most 2 points of intersection, and never 4 as predicted by Bézout’s theorem. This follows from the fact that the quadratic term in the equation of all circles is always the same, namely
x2 + y2, so that subtracting the equation of one circle from that of the other gives a linear equation, and therefore the intersection of two circles is the same thing as the intersection of a circle and a line. Moreover, if the circles are not tangent, their multiplicity of intersection is 1 at each point of intersection. To understand what lies behind this failure of Bézout’s theorem, write the equation of the circle (x − a)2 + (y − b)2 = r 2 in homogeneous coordinates by setting x = ξ/ζ and y = η/ζ. We get the equation (ξ − aζ )2 + (η − bζ )2 = r 2ζ 2, from which we see that the circle intersects the line at inﬁnity ζ = 0 in the points ξ 2 + η2 = 0, that is, in the two circular points at inﬁnity (1, ±i, 0). Thus all circles have the two points (1, ±i, 0) at inﬁnity in common. Taken together with the two ﬁnite points of intersection, we thus get 4 points of intersection, in agreement with Bézout’s theorem. This type of phenomenon motivates passing from the afﬁne to the projective plane. Recall that a point of the projective plane P2 is determined by 3 elements (ξ, η, ζ ) of the ﬁeld k, not all simultaneously zero. Two triples (ξ, η, ζ ) and (ξ, η, ζ ) determine the same point if there exists λ ∈ k with λ = 0 such that ξ = λξ, η = λη and ζ = λζ. Any triple (ξ, η, ζ ) deﬁning a point P is called a set of homogeneous coordinates of P, and we write P = (ξ : η : ζ ). There is an inclusion A2 ⊂ P2 which sends (x, y) ∈ A2 to (x : y : 1). We get in this way all points with ζ = 0: a point (ξ : η : ζ ) ∈ P2 with �
� = 0 corresponds to the point (ξ/ζ, η/ζ ) ∈ A2. The points of the complementary set ζ = 0 are called points at inﬁnity. This notion is related to the choice of the coordinate ζ. In fact, P2 contains 3 sets that are copies of the afﬁne plane in this way: A2 1 (given by ξ = 0), A2 3 (given by ζ = 0). These intersect, of course: if a point 2 (given by η = 0), and A2 18 1 Basic Notions 3 has coordinates x = ξ/ζ, y = η/ζ and η = 0 then in A2 P ∈ A2 2 the same point has coordinates x = ξ/η, y = ζ /η, so that x = x/y, y = 1/y; if ξ = 0 then in A2 1 it has coordinates x = η/ξ, y = ζ /ξ, so that x = y/x, y = 1/x. Every point P ∈ P2 is contained in at least one of the pieces A2 3, and can be written down in the afﬁne coordinates of that piece. 2 or A2 1, A2 An algebraic curve in P2, or a projective algebraic plane curve is deﬁned in homogeneous coordinates by an equation F (ξ, η, ζ ) = 0, where F is a homogeneous polynomial. Then whether F (ξ, η, ζ ) = 0 holds or not is independent of the choice of the homogeneous coordinates of a point; that is, it is preserved on passing from ξ, η, ζ to ξ = λξ, η = λη, ζ = λζ with λ = 0. A homogeneous polynomial is also called a form. An afﬁne algebraic curve of degree n with equation f (x, y) = 0 deﬁnes a homogeneous polynomial F (ξ, η, ζ ) = ζ nf (ξ/ζ, η/ζ ), and hence a projective curve with equation F (ξ, η, ζ
) = 0. It is easy to see that intersecting this curve with the afﬁne plane A2 3 gives us the original afﬁne curve, to which it therefore only adds points at inﬁnity with ζ = 0. If the equation of the projective curve is F (ξ, η, ζ ) = 0, then that of the corresponding afﬁne curve is f (x, y) = 0, where f (x, y) = F (x, y, 1). Since every point P ∈ P2 is contained in one of the afﬁne sets A2 1, A2 3, we can use this correspondence to write out the properties of curves, deﬁned above for afﬁne curves, in terms of homogeneous coordinates. We do this now for the notions of tangent line, singular point and inﬂexion point of an algebraic curve. We always assume that P ∈ A2 3. 2 or A2 In afﬁne coordinates, the equation of the tangent is ∂f ∂x (P )(x − α) + ∂f ∂y (P )(y − β) = 0. By assumption f (x, y) = F (x, y, 1), where F (ξ, η, ζ ) = 0 is the homogeneous equation of our curve. Hence writing F x etc. for the partial derivatives, we get ∂f/∂x = F y(x, y, 1), and by the well-known theorem of Euler on homogeneous functions, we have x(x, y, 1) and ∂f/∂y = F ξ ξ + F F ηη + F Since P = (α : β : 1) is a point of the curve, F the equation of the tangent is F coordinates ξ (P )x + F ζ ζ = nF. ξ (P )α + F η(P )y + F η(P )β + F ζ (P ) = 0, so that ζ (P ) = 0, or in homogeneous ξ (P )ξ + F F η(P )η + F ζ (P )ζ = 0. The conditions in afﬁ
ne coordinates for a singular point are f = f = 0. x = F = 0, and by Euler’s theorem, = 0. If the characteristic of the ﬁeld k is 0 then it is enough to Hence in homogeneous coordinates F ξ since ζ = 1, also F ζ require the conditions F ζ (P ) = 0, since then also F (P ) = 0. ξ (P ) = F η( The condition deﬁning an inﬂexion point is given by the relation (1.15). Here again f (x, y) = F (x, y, 1), so that f x, f = x F yy. From now on, in the homogeneous polynomial F we write ξ for x and η xx, f xy, xx yy xy y 1 Algebraic Curves in the Plane 19 for y. We substitute these expressions in the determinant of (1.15), and use Euler’s theorem ξ ξ ξ + F F ξ ηξ + F F ξ ξ + F F ξ ηη + F ηηη + F ηη + F ξ ζ ζ = (n − 1)F ξ, ζ ηζ = (n − 1)F η, ζ ζ = nF. Multiply the last column of our determinant by (n − 1), and subtract from it ξ times the ﬁrst column and η times the second. Using the above identities and recalling that F (P ) = 0, we get the determinant ξ ζ ξ η F ξ ξ F F ηη P ). Now perform the same operation on the rows of the determinant. The condition for P to be an inﬂexion point then takes the form ξ ζ F ξ ξ F F ηξ F ζ ξ F F ξ η F ηη F ζ η F ηζ ζ ζ (P ) = 0. (1.16) The determinant on the left-hand side of (1.16) is called the Hessian form of F, and denoted by H (F ). We now proceed to considering rational functions
. Making the substitution x = ξ/ζ, y = η/ζ and clearing denominators, we can rewrite a rational function f = p(x, y)/q(x, y) on A2 3 in the form P (ξ, η, ζ )/Q(ξ, η, ζ ), where P and Q are homogeneous polynomials of the same degree. Hence its value at a point (ξ : η : ζ ) does not change on multiplying the homogeneous coordinates through by a common multiple, and hence f can be viewed as a partially deﬁned function on P2. → A2 3 deﬁned by (x, y) → (u(x, y), v(x, y)), we Given a rational map ϕ : A2 3 ﬁrst rewrite it, as just explained, in the form U (ξ, η, ζ ) R(ξ, η, ζ ), V (ξ, η, ζ ) S(ξ, η, ζ ), where U, V, R, S are homogeneous polynomials, with deg U = deg R and deg V = deg S. Next we put the two components over a common denominator, that is, in the form (A/C, B/C), with deg A = deg B = deg C. Finally, introducing homogeneous coordinates ξ /ζ = A/C, η/ζ = B/C, we write the map in the form (ξ : η : ζ ) → A(ξ : η : ζ ) : B(ξ : η : ζ ) : C(ξ : η : ζ ), where A, B, C are homogeneous polynomials of the same degree. Now ϕ is naturally a rational map P2 → P2. The map is regular at a point P if one of A, B, C does 20 1 Basic Notions not vanish at P. Studying properties related to points P in the afﬁne set A2 3, say, we can divide each of A, B, C by ζ n, where n is their common degree, and write the map in the form (x, y) → (u(x, y), v(
x, y), w(x, y)), where u, v and w are polynomials. This map is regular at P if the 3 polynomials do not vanish simultaneously at P. As a ﬁrst illustration we prove the following important result. Theorem 1.2 A rational map from a projective plane curve C to P2 is regular at every nonsingular point of C (see Section 1.5 for the deﬁnition). Proof Suppose that the nonsingular point P is in the afﬁne piece A2 3 with coordinates denoted by x, y. We write the map as above in the form (x, y) → (u0 : u1 : u2) where u0, u1, u2 are polynomials, and apply Theorem 1.1 to these. Restricting the ui to C, we can write them in the form ui = t ki vi, where t is a local parameter, vi(P ) = 0 and ki ≥ 0 for i = 0, 1, 2. Suppose that k0, say, is the smallest of the numbers k0, k1, k2. Then the same map can be rewritten in the form (x, y) → (v0 : t k1−k0 v1 : t k2−k0v2), with k1 − k0 ≥ 0, k2 − k0 ≥ 0, and v0(P ) = 0. It follows that it is regular at P. The theorem is proved. Corollary A birational map between nonsingular projective plane curves is regular at every point, and is a one-to-one correspondence. As an example, consider a birational map of the projective line to itself. Just as with any rational map, this can be written as a rational function x → p(x)/q(x), with p(x), q(x) ∈ k[x] (here we assume that x is a coordinate on our line, for example the line given by y = 0). The points that map to a given point α are those for which p(x)/q(x) = α, that is, p(x) − αq(x) = 0. Hence from the fact that the map is birational, it follows that p and q are linear, that is, the map is of the form x → (
ax + b)/(cx + d) with ad − bc = 0. As a consequence, we get that a birational map of the line to itself has at most two ﬁxed points, the roots of the equation x(cx + d) = ax + b. Now consider the elliptic curve given by (1.10), and assume that 4p3 + 27q2 = 0. All its ﬁnite points are nonsingular. Passing to homogeneous coordinates, we can write its equation in the form η2ζ = ξ 3 + pξ ζ 2 + qζ 3. Hence it has a unique point on the line at inﬁnity ζ = 0, namely the point o = (0 : 1 : 0). Dividing through by η3 we write the equation of the curve in the form v = u3 + puv2 + qv3, in coordinates u, v, where u = ξ/η and v = ζ /η. The point o = (0, 0) in these coordinates is also nonsingular. Hence our curve is nonsingular. The map (x, y) → (x, −y) is obviously a birational map of the curve to itself. Its ﬁxed points in the ﬁnite part of the plane are the points with y = 0, x3 + px + q = 0, that is, there are 3 such points. The point o is also a ﬁxed point, since u = x/y, v = 1/y, and in coordinates u, v, the map is written (u, v) → (−u, −v). We have constructed on an elliptic curve an automorphism having 4 ﬁxed points. It follows from this that an elliptic curve is not birational to a line, that is, is not rational. This shows that the problem of birational classiﬁcation of curves is not trivial: not all curves are birational to one another. 1 Algebraic Curves in the Plane 21 Passing to projective curves is the ﬁnal amendment required in the statement of Bézout’s theorem. One version of this is as follows: Theorem Let X and Y be projective curves, with X nonsingular and not contained in Y.
Then the sum of the multiplicities of intersection of X and Y at all points of X ∩ Y equals the product of the degrees of X and Y. We will prove this theorem and a series of generalisations in a later section (Section 2.2, Chapter 3 and Section 2.1, Chapter 4). Here we verify the two simplest cases, when X is a line or a conic. Let X be a line. By Lemma of Section 1.1, X and Y have a ﬁnite number of points of intersection. We choose a convenient coordinate system, so that the line ζ = 0 does not pass through the points of intersection, and is not equal to X, and η = 0 is the line X. Then the points of intersection of X and Y are contained in the afﬁne plane with coordinates x = ξ/ζ, y = η/ζ, and the equation of X is y = 0. Let f (x, y) = 0 be the equation of the curve Y and f = f0 + f1(x, y) + · · · + fn(x, y) its expression as a sum of homogeneous polynomials. The point (1 : 0 : 0) is not contained in Y by the choice of the coordinate system, and hence fn(1, 0) = 0, that is, f contains the term axn with a = 0. Hence f (x, 0), the restriction of f to X, has degree n. The function x − α is a local parameter of X at the point x = α, and the multiplicity of intersection of X and Y at this point equals the multiplicity of the root x = α of the polynomial f (x, 0). Therefore the sum of these multiplicities equals n. Let X be a conic. Take any point P ∈ X with P /∈ Y, and choose coordinates so that ζ = 0 is the tangent line to X at P, and ξ = 0 some other line through P. An easy calculation in coordinates shows that X is a parabola in the afﬁne plane with coordinates x = ξ/ζ, y = η/ζ (since it touches the line at inﬁnity), with equation y = px2 +qx +r and p = 0. As before, f = f0 +
· · ·+fn(x, y), and now fn(0, 1) = 0, that is, f (x, y) contains the term ayn with a = 0. The conic X has no other points of intersection with the line ζ = 0 except P, and hence all the points of intersection of X and Y are contained in the ﬁnite part of the plane. At any point with x = α the function x − α is a local parameter on X, and the multiplicity of intersection of X and Y at this point is equal to the multiplicity of the root x = α of the polynomial f (x, px2 + qx + r). Since f (x, y) contains the term ayn with a = 0, the degree of f (x, px2 + qx + r) is 2n, so that the sum of multiplicities of all the points of intersection equals 2n. This proves the theorem in the case X is a line or conic. Already this simple particular case of Bézout’s theorem has beautiful geometric applications. One of these is the proof of Pascal’s theorem, which asserts that for a hexagon inscribed in a conic, the 3 points of intersection of pairs of opposite sides are collinear. Let l1 and m1, l2 and m2, l3 and m3 be linear forms that are the equations of the opposite sides of a hexagon (see Figure 6). Consider the cubic with the equation fλ = l1l2l3 + λm1m2m3 where λ is an arbitrary parameter. This has six points of intersection with the conic, the vertexes of the hexagon. Moreover, we can choose the value of λ so that fλ(P ) = 0 for 22 Figure 6 Pascal’s theorem 1 Basic Notions any given point P ∈ X, distinct from these 6 points of intersection. We get a cubic fλ having 7 points of intersection with a conic X, and by Bézout’s theorem this must decompose as the conic X plus a line L. This line L must contain the points of intersection l1 ∩ m1, l2 ∩ m2 and l3 ∩ m3. (This proof is due to Plücker.) 1.7 Exercises to Section 1 1 Find a characterisation in real
terms of the line through the points of intersection of two circles in the case that both these points are complex. Prove that it is the locus of points having the same power with respect to both circles. (The power of a point with respect to a circle is the square of the distance between it and the points of tangency of the tangent lines to the circle.) 2 Which rational functions p(x)/q(x) are regular at the point at inﬁnity of P1? What order of zero do they have there? 3 Prove that an irreducible cubic curve has at most one singular point, and that the multiplicity of a singular point is 2. If the singularity is a node then the cubic is projectively equivalent to the curve in (1.2); and if a cusp then to the curve y2 = x3. 4 What is the maximum multiplicity of intersection of two nonsingular conics at a common point? 5 Prove that if the ground ﬁeld has characteristic p then every line through the origin is a tangent line to the curve y = xp+1. Prove that over a ﬁeld of characteristic 0, there are at most a ﬁnite number of lines through a given point tangent to a given irreducible curve. 6 Prove that the sum of multiplicities of two singular points of an irreducible curve of degree n is at most n, and the sum of multiplicities of any 5 points is at most 2n. 2 Closed Subsets of Afﬁne Space 23 7 Prove that for any two distinct points of an irreducible curve there exists a rational function that is regular at both, and takes the value 0 at one and 1 at the other. 8 Prove that for any nonsingular points P1,..., Pr of an irreducible curve and numbers m1,..., mr ≥ 0 there exists a rational function that is regular at all these points, and has a zero of multiplicity mi at Pi. 9 For what values of m is the cubic x3 0 Find its inﬂexion points. + x3 1 + x3 2 + mx0x1x2 = 0 in P2 nonsingular? 10 Find all the automorphisms of the curve of (1.2). 11 Prove that on the projective line
and on a conic of P2, a rational function that is regular at every point is a constant. 12 Give an interpretation of Pascal’s theorem in the case that pairs of vertexes of the hexagon coincide, and the lines joining them become tangents. 2 Closed Subsets of Afﬁne Space Throughout what follows, we work with a ﬁxed algebraically closed ﬁeld k, which we call the ground ﬁeld. 2.1 Deﬁnition of Closed Subsets At different stages of the development of algebraic geometry, there have been changing views on the basic object of study, that is, on the question of what is the “natural deﬁnition” of an algebraic variety; the objects considered to be most basic have been projective or quasiprojective varieties, abstract algebraic varieties, schemes or algebraic spaces. In this book, we consider algebraic geometry in a gradually increasing degree of generality. The most general notion considered in the ﬁrst chapters, embracing all the algebraic varieties studied here, is that of quasiprojective variety. In the ﬁnal chapters this role will be taken by schemes. At present we deﬁne a class of algebraic varieties that will play a foundational role in all the subsequent deﬁnitions. Since the word variety will be reserved for the more general notions, we use a different word here. We write An for the n-dimensional afﬁne space over the ﬁeld k. Thus its points are of the form α = (α1,..., αn) with αi ∈ k. Deﬁnition A closed subset of An is a subset X ⊂ An consisting of all common zeros of a ﬁnite number of polynomials with coefﬁcients in k. We will sometimes say simply closed set for brevity. 24 1 Basic Notions From now on we will write F (T ) to denote a polynomial in n variables, allowing T to stand for the set of variables T1,..., Tn. If a closed set X consists of all common zeros of polynomials F1(T ),..., Fm(T ), then we refer to F1(T ) = · · · = F
m(T ) = 0 as the equations of the set X. A set X deﬁned by an inﬁnite system of equations Fα(T ) = 0 is also closed. Indeed, the ideal A of the polynomial ring in T1,..., Tn generated by all the polynomials Fα(T ) is ﬁnitely generated (the Hilbert Basis Theorem, see Atiyah and Macdonald [8, Theorem 7.5]), that is, A = (G1,..., Gm). One checks easily that X is deﬁned by the system of equations G1 = · · · = Gm = 0. It follows from this that the intersection of any number of closed sets is closed. Xα, Indeed, if Xα are closed sets, then to get a system of equations deﬁning X = we need only take the union of the systems deﬁning all the Xα. The union of a ﬁnite number of closed sets is again closed. It is obviously enough to check this for two sets. If X = X1 ∪ X2, where X1 is deﬁned by the system of equations Fi(T ) = 0 for i = 1,..., m and X2 by Gj (T ) = 0 for j = 1,..., l then it is easy to check that X is deﬁned by the system Fi(T )Gj (T ) = 0 for i = 1,..., m and j = 1,..., l. Let X ⊂ An be a closed subset of afﬁne space. We say that a set U ⊂ X is open if its complement X \ U is closed. Any open set U x is called a neighbourhood of x. The intersection of all the closed subsets of X containing a given subset M ⊂ X is closed. It is called the closure of M and denoted by M. A subset is dense in X if M = X. This means that M is not contained in any closed subset Y X. Example 1.1 The whole afﬁne space An is closed, since it is deﬁned by the empty set of equations, or by 0 = 0. Example 1.2 The subset X ⊂ A1
consisting of all points except 0 is not closed: every polynomial F (T ) that vanishes at all T = 0 must be identically 0. Example 1.3 Let us determine all the closed subsets X ⊂ A1. Such a set is given by a system of equations F1(T ) = · · · = Fm(T ) = 0 in one variable T. If all the Fi are identically 0 then X = A1. If the Fi don’t have any common factor, then they don’t have any common roots, and X does not contain any points. If the highest common factors of all the Fi is D(T ) then D(T ) = (T − α1) · · · (T − αn) and X consists of the ﬁnitely many points T = α1,..., T = αn. Example 1.4 Let us determine all the closed subsets X ⊂ A2. A closed subset is given by a system of equations F1(T ) = · · · = Fm(T ) = 0, (1.17) where now T = (T1, T2). If all the Fi are identically 0 then X = A2. Suppose this is not the case. If the polynomials F1,..., Fm do not have a common factor then, as follows from Lemma of Section 1.1, the system (1.17) has only a ﬁnite set of solutions (possibly empty). Finally, suppose that the highest common factor of all 2 Closed Subsets of Afﬁne Space 25 the Fi(T ) is D(T ). Then Fi(T ) = D(T )Gi(T ), where now the polynomials Gi(T ) do not have a common factor. Obviously then X = X1 ∪ X2 where X1 is given by G1(T ) = · · · = Gm(T ) = 0 and X2 is given by the single equations D(T ) = 0. As we have seen, X1 is a ﬁnite set. The closed sets deﬁned in A2 by one equation are the algebraic plane curves. Thus a closed set X ⊂ A2 either consists of a ﬁnite set of points (possibly empty), or the union of an algebra
ic plane curve and a ﬁnite set of points, or the whole of A2. Example 1.5 If α ∈ Ar is the point with coordinates (α1,..., αr ) and β ∈ As the point with coordinates (β1,..., βs), we take α, β into the point (α, β) ∈ Ar+s with coordinates (α1,..., αr, β1,..., βs). Thus we identify Ar+s as the set of pairs (α, β) with α ∈ Ar and β ∈ As. Let X ⊂ Ar and Y ⊂ As be closed sets. The set of pairs (x, y) ∈ Ar+s with x ∈ X and y ∈ Y is called the product of X and Y, and denoted by X × Y. This is again a closed set. Indeed, if X is given by Fi(T ) = 0 and Y by Gj (U ) = 0 then X × Y ⊂ Ar+s is deﬁned by Fi(T ) = Gj (U ) = 0. Example 1.6 A set X ⊂ An deﬁned by one equation F (T1,..., Tn) = 0 is called a hypersurface. 2.2 Regular Functions on a Closed Subset Let X be a closed set in the afﬁne space An over the ground ﬁeld k. Deﬁnition A function f deﬁned on X with values in k is regular if there exists a polynomial F (T ) with coefﬁcients in k such that f (x) = F (x) for all x ∈ X. If f is a given function, the polynomial F is in general not uniquely determined. We can add to F any polynomial entering in the system of equations of X without altering f. The set of all regular functions on a given closed set X forms a ring and an algebra over k; the operations of addition, multiplication and scalar multiplication by elements of k, are deﬁned as in analysis, by performing the operations on the value of the functions at each point x ∈ X.The ring obtained in this way is denoted by k[X] and is called the coordinate ring of
X. We write k[T ] for the polynomial ring with coefﬁcients in k in variables T1,..., Tn. We can obviously associate with each polynomial F ∈ k[T ] a function f ∈ k[X], by viewing F as a function on the set of points of X; in this way we get a homomorphism from k[T ] to k[X]. The kernel of this homomorphism consists of all polynomials F ∈ k[T ] that take the value 0 at every point x ∈ X. This is an ideal of k[T ], just as the kernel of any ring homomorphism; it is called the ideal of the closed set X, and denoted by AX. Obviously Thus k[X] is determined by the ideal AX ⊂ k[T ]. k[X] = k[T ]/AX. 26 1 Basic Notions Example 1.7 If X is a point then k[X] = k. Example 1.8 If X = An then AX = 0 and k[X] = k[T ]. Example 1.9 Let X ⊂ A2 be given by the equation T1T2 = 1. Then k[X] = ], and it consists of all the rational functions in T1 of the form G(T1)/T n k[T1, T 1 with G(T1) a polynomial and n ≥ 0. −1 1 Example 1.10 We prove that if X and Y are any closed sets then k[X × Y ] = k[X] ⊗k k[Y ]. Deﬁne a homomorphism ϕ : k[X] ⊗k k[Y ] → k[X × Y ] by the condition fi ⊗ gi (x, y) = ϕ i i fi(x)gi(y). The right-hand side is obviously a regular functions on X × Y, and it is clear that ϕ is onto, since, in the notation of Example 1.5, the functions αi and βj are contained in the image of ϕ, and these generate k[X × Y ]. To prove that ϕ is one-to-one, it is enough to check that if {fi} are linearly independent in k[X] and {gj } in k[Y ] then
{fi ⊗ gj } are linearly independent in k[X × Y ]. Now an equality cij fi(x)gj (y) = 0 implies the relation j cij gj (y) = 0 for any ﬁxed y, and in turn that cij = 0. i,j Since k[X] is a homomorphic image of the polynomial ring k[T ], it satisﬁes the Hilbert basis theorem: any ideal of k[X] is ﬁnitely generated. It also satisﬁes the following analogue of the Nullstellensatz (Proposition A.9): if a function f ∈ k[X] is zero at every point x ∈ X at which functions g1,..., gm vanish then f r ∈ (g1,..., gm) for some r > 0. Indeed, suppose that f is given by a polynomial F (T ), the gi by polynomials Gi(T ), and let Fj = 0 for j = 1,..., l be the equations of X. Then F (T ) vanishes at all points α ∈ An at which all the polynomials G1,..., Gm, F1,..., Fl vanish; for since Fj (α) = 0 it follows that α ∈ X, and then by assumption F (α) = 0. Applying the Nullstellensatz in the polynomial ring we deduce that F r ∈ (G1,..., Gm, F1,..., Fl) for some r > 0, and hence f r ∈ (g1,..., gm) in k[X]. How is the ideal AX of a closed set X related to a system F1 = · · · = Fm = 0 of deﬁning equations of X? Clearly Fi ∈ AX by deﬁnition of AX, and hence (F1,..., Fm) ⊂ AX; however, it’s not always true that (F1,..., Fm) = AX. For example, if X ⊂ A1 is deﬁned by the equation T 2 then it consists just of the point T = 0, so that
AX consists of all polynomials with no constant term. That is, AX = (T ), whereas (F1,..., Fm) = (T 2). We can however always deﬁne the same closed set X by a system of equation G1 = · · · = Gl = 0 in such a way that AX = (G1,..., Gl). For this it is enough to recall that any ideal of k[T ] is ﬁnitely generated. Let G1,..., Gl be a basis of the ideal AX, that is, AX = (G1,..., Gl). 2 Closed Subsets of Afﬁne Space 27 Then obviously the equations G1 = · · · = Gl = 0 deﬁne the same set X and have the required property. It is sometimes even convenient to consider a closed set as deﬁned by the inﬁnite system of equations F = 0 for all polynomials F ∈ AX. Indeed, if (F1,..., Fm) = AX then these equations are all consequences of F1 = · · · = Fm = 0. Relations between closed subsets are often reﬂected in their ideals. For example, if X and Y are closed sets in the afﬁne space An then X ⊃ Y if and only if AX ⊂ AY. It follows from this that with any closed subset Y contained in X we can associate the ideal aY of k[X], consisting of the images under the homomorphism k[T ] → k[X] of polynomials F ∈ AY. Conversely, any ideal a of k[X] deﬁnes an ideal A in k[T ], consisting of all inverse images under k[T ] → k[X] of elements of a. Clearly A ⊃ AX. The equations F = 0 for all F ∈ A deﬁne the closed set Y ⊂ X. It follows from the Nullstellensatz that Y is the empty set if and only if aY = k[X]. The ideal aY ⊂ k[X] can alternatively be described as the set of all functions f ∈ k[X] that vanish at all points of the subset Y. In particular, each point x ∈ X is
a closed subset, and hence deﬁnes an ideal mx ⊂ k[X]. By deﬁnition this ideal is the kernel of the homomorphism k[X] → k that takes a function f ∈ k[X] to its value f (x) at x. Since k[X]/mx = k is a ﬁeld, the ideal mx is maximal. Conversely, every maximal ideal m ⊂ k[X] corresponds in this way to some point x ∈ X. Indeed, it deﬁnes a closed subset Y ⊂ X; for any point y ∈ Y we have my ⊃ m, and then my = m since m is maximal. For u ∈ k[X] the set of points x ∈ X at which u(x) = 0 is closed; it is denoted by V (u), and called a hypersurface in X. 2.3 Regular Maps Let X ⊂ An and Y ⊂ Am be closed subsets. Deﬁnition A map f : X → Y is regular if there exist m regular functions f1,..., fm on X such that f (x) = (f1(x),..., fm(x)) for all x ∈ X. Thus any regular map f : X → Am is given by m functions f1,..., fm ∈ k[X]; in order to know that this maps into the closed subset Y ⊂ Am, it is obviously enough to check that f1,..., fm as elements of k[X] satisfy the equations of Y, that is G(f1,..., fm) = 0 ∈ k[X] for all G ∈ AY. Example 1.11 A regular function on X is exactly the same thing as a regular map X → A1. Example 1.12 A linear map An → Am is a regular map. Example 1.13 The projection map (x, y) → x deﬁnes a regular map of the curve deﬁned by xy = 1 to A1. 28 1 Basic Notions Example 1.14 The preceding example can be generalised as follow: let X ⊂ An be a closed subset and F a regular function on X. Consider the subset X ⊂
X × A1 deﬁned by the equation Tn+1F (T1,..., Tn) = 1. The projection ϕ(x1,..., xn+1) = (x1,..., xn) deﬁnes a regular map ϕ : X → X. Example 1.15 The map f (t) = (t 2, t 3) is a regular map of the line A1 to the curve given by y2 = x3. Example 1.16 (The zeta function of a variety over Fp) We give an example that is very important for number theory. Suppose that the coefﬁcients of the equations Fi(T ) of a closed subset X ⊂ An belong to the ﬁeld Fp with p elements, where p is a prime number. As we said in Section 1.1, the points of X with coordinates in Fp correspond to solutions of the system of congruences Fi(T ) ≡ 0 mod p. Consider the map ϕ : An → An deﬁned by ϕ(α1,..., αn) = 1,..., αp αp n. This is obviously a regular map. The important thing is that it takes X ⊂ An to itself. Indeed, if α ∈ X, that is, Fi(α) = 0, then since Fi(T ) ∈ Fp[T ], it follows from properties of ﬁelds of characteristic p that Fi(αp n ) = (Fi(α1,..., αn))p = 0. The map ϕ : X → X obtained in this way is called the Frobenius map. Its signiﬁcance is that the points of X with coordinates in Fp are characterised among all points of X as the ﬁxed points of ϕ. Indeed, the solutions of the equation αp = αi are exactly i all the elements of Fp. 1,..., αp In exactly the same way, the elements α ∈ Fpr of the ﬁeld with pr elements are characterised as the solutions of αpr = α, and hence the points x ∈ X with coordinates in Fpr are the ﬁxed points of the map ϕr. For each
r, write νr for the number of points x ∈ X with coordinates in Fpr. To get a better overall view of the set of numbers νr, we consider the generating function PX(t) = ∞ r=1 νr t r. A deep general theorem asserts that this function is always a rational function of t (for a fairly elementary proof, see Koblitz [49, Chapter V]). In this way the function PX(t) gives an expression in ﬁnite terms for the inﬁnite sequence of numbers νr. The function PX(t) associated with the closed set X has some properties analogous to those of the Riemann zeta function. To express these, note that if x ∈ X is a point whose coordinates are in Fpr and generate this ﬁeld, then X contains all the points ϕi(x) for i = 1,..., r, and these are all distinct. We call a set ξ = {ϕi(x)} of this form a cycle, and the number r of points of ξ the degree of ξ, denoted deg ξ. Now we can group together all the νr points x ∈ X with coordinates in Fpr into cycles. The coordinates of any of these points generate some subﬁeld F pd ⊂ Fpr, and 2 Closed Subsets of Afﬁne Space 29 it is known that d \| r (see for example van der Waerden [76, Ex. 6.23 of Section 6.7 (Ex. 1 of Section 43)]). We get a formula νr = d\|r dμd, where μd is the number of cycles of degree d, hence PX(t) = ∞ dμd t r = ∞ ∞ dμd t md = r=1 d\|r d=1 m=1 ∞ d=1 μd dt d 1 − t d. We introduce the function ZX(t) = ξ 1 1 − t deg ξ, (1.18) (1.19) where the product runs over all cycles ξ. Then the formula (1.18) can obviously be rewritten as PX(t) = Z X(t) ZX(t) t. Equation (1.19) is analogous to the
Euler product for the Riemann zeta function. To emphasise this analogy we set pdeg ξ = N(ξ ) and t = p−s. Then (1.19) takes the form ZX(t) = ζX(s) = ξ 1 1 − N(ξ )−s. This function (either ZX(t) or ζX(s)) is called the zeta function of X. We now ﬁnd out how a regular map acts on the ring of regular functions on a closed set. We start with a remark concerning arbitrary maps between sets. If f : X → Y is a map from a set X to a set Y then we can associate with every function u on Y (taking values in an arbitrary set Z) a function v on X by setting v(x) = u(f (x)). Obviously the map v : X → Z is the composite of f : X → Y and u : Y → Z. We set v = f ∗(u), and call it the pullback of u. We get in this way a map f ∗ from functions on Y to functions on X. Now suppose that f colonX → Y is a regular map; then f ∗ takes regular functions on Y into regular functions on X. Indeed, if u is given by a polynomial function G(T1,..., Tn) and f by polynomials F1,..., Fm then v = f ∗(u) is obtained simply by substituting Fi for Ti in G, so that v is given by the polynomial G(F1,..., Fm). Moreover, regular maps can be characterised as the maps that take regular functions into regular functions. Indeed, suppose that a map f : X → Y of closed set has the property that for any regular function u on Y the function f ∗(u) on X is again regular. Then this applies in particular to the functions ti deﬁned by the coordinates Ti on Y for i = 1,..., m; thus the functions f ∗(ti) are regular on X. But this just means that f is a regular map. 30 1 Basic Notions We have seen that if f is regular then the pullback of functions deﬁnes a map f ∗ : k[Y ] → k[X]. It
follows easily from the deﬁnition of f ∗ that it is a homomorphism of k-algebras. We show that, conversely, every algebra homomorphism ϕ : k[Y ] → k[X] is of the form ϕ = f ∗ for some regular map f : X → Y. Let t1,..., tm be coordinates in the ambient space Am of Y, viewed as functions on Y. Obviously ti ∈ k[Y ], and hence ϕ(ti) ∈ k[X]. Set ϕ(ti) = si and consider the map f given by the formula f (x) = (s1(x),..., sm(x)). This is of course a regular map. We prove that f (X) ⊂ Y. Indeed, if H ∈ AY then H (t1,..., tm) = 0 in k[Y ], hence also ϕ(H ) = 0 on X. Let x ∈ X; then H (f (x)) = ϕ(H )(x) = 0, and therefore f (x) ∈ Y. Deﬁnition A regular map f : X → Y of closed sets is an isomorphism if it has an inverse, that is, if there exists a regular map g : Y → X such that f ◦ g = 1 and g ◦ f = 1. In this case we say that X and Y are isomorphic. An isomorphism is obviously a one-to-one correspondence. It follows from what we have said that if f is an isomorphism then f ∗ : k[Y ] → k[X] is an isomorphism of algebras. It is easy to see that the converse is also true; in other words, closed sets are isomorphic if and only if their rings of regular functions are isomorphic over k. The facts we have just proved show that X → k[X] deﬁnes an equivalence of categories between closed subsets of afﬁne spaces (with regular maps between them) and a certain subcategory of the category of commutative algebras over k (with algebra homomorphisms). What is this subcategory, that is, which algebras are of the form k[X]? Theorem 1.3 An algebra
A over a ﬁeld k is isomorphic to a coordinate ring k[X] of some closed subset X if and only if A has no nilpotents (that is f m = 0 implies that f = 0 for f ∈ A) and is ﬁnitely generated as an algebra over k. Proof These conditions are all obviously necessary. If an algebra A is generated by ﬁnitely many elements t1,..., tn then A ∼= k[T1,..., Tn]/A, where A is an ideal of the polynomial ring k[T1,..., Tn]. Suppose that A = (F1,..., Fm), and consider the closed set X ⊂ An deﬁned by the equations F1 = · · · = Fm = 0; we prove that AX = A, from which it will follow that k[X] ∼= k[T1,..., Tn]/A ∼= A. If F ∈ AX then F r ∈ A for some r > 0 by the Nullstellensatz. Since A has no nilpotents, also F ∈ A. Thus AX ⊂ A, and since obviously A ⊂ AX, we have AX = A. The theorem is proved. Example 1.17 The generalised parabola, deﬁned by the equation y = xk is isomorphic to the line, and the maps f (x, y) = x and g(t) = (t, t k) deﬁne an isomorphism. Example 1.18 The projection f (x, y) = x of the hyperbola xy = 1 to the x-axis is not an isomorphism, since the map is not a one-to-one correspondence: the hyperbola does not contain any point (x, y) for which f (x, y) = 0. Compare also Exercise 4. 2 Closed Subsets of Afﬁne Space 31 Example 1.19 The map f (t) = (t 2, t 3) of the line to the curve deﬁned by y2 = x3 is easily seen to be a one-to-one correspondence. However, it is not an isomorphism, since the inverse map is of the form g(x, y
) = y/x, and the function y/x is not regular at the origin. (See Exercise 5.) Example 1.20 Let X and Y ⊂ An be closed sets. Consider X × Y ⊂ A2n as in Example 1.5, and the linear subspace Δ ⊂ A2n deﬁned by equations t1 = u1,..., tn = un, called the diagonal. Consider the map that sends each point z ∈ X ∩ Y to ϕ(z) = (z, z) ∈ A2n, which is obviously a point of X × Y ∩ Δ. It is easy to check that the map obtained in this way is an isomorphism from X ∩ Y to X × Y ∩ Δ. Using this, we can always reduce the study of the intersection of two closed sets to considering the intersection of a different closed set with a linear subspace. Example 1.21 Let X be a closed set and G a ﬁnite group of automorphisms of X. Suppose that the characteristic of the ﬁeld k does not divide the order N of G. Set A = k[X], and let AG be the subalgebra of invariants of G in A, that is, AG = {f ∈ A \| g∗(f ) = f for allg ∈ G}. According to Proposition A.6, the algebra AG is ﬁnitely generated over k. From Theorem 1.3 it follows that there exists a closed set Y such that AG ∼= k[Y ], and a regular map ϕ : X → Y such that ϕ∗(k[Y ]) = AG. This set Y is called the quotient variety or quotient space of X by the action of G, and is written X/G. Given two points x1, x2 ∈ X, there exists g ∈ G such that x2 = g(x1) if and only if ϕ(x1) = ϕ(x2). Indeed, if x2 = g(x1) then f (x2) = f (x1) for every f ∈ k[X]G = k[Y ], and hence ϕ(x1) = ϕ(x2). Conversely, if x2 = g(x1) then we must take a function f ∈
k[X] such that f (g(x2)) = 1, f (g(x1)) = 0 for all g ∈ G. Then the symmetrised function S(f ) (see Section 4, Appendix) is G-invariant and satisﬁes S(f )(x2) = 1 and S(f )(x1) = 0, and hence ϕ(x2) = ϕ(x1). Thus X/G parametrises the orbits {g(x) \| g ∈ G} of G acting on X. In what follows we will mainly be interested in notions and properties of closed sets invariant under isomorphism. The system of equations deﬁning a set is clearly not a notion of this kind; two sets X and Y can be isomorphic although given by different systems of equations in different spaces An. Thus it would be natural to try to give an intrinsic deﬁnition of a closed set independent of its realisation in some afﬁne space; a deﬁnition of this kind will be given in Chapters 5–6 in connection with the notion of a scheme. Now we determine when a homomorphism f ∗ : k[Y ] → k[X] corresponding to a regular map f : X → Y has no kernel, that is, when f ∗ is an isomorphic inclusion k[Y ] → k[X]. For u ∈ k[Y ], let’s see when f ∗(u) = 0. This means that u(f (x)) = 0 for all x ∈ X. In other words, u vanishes at all points of the image f (X) of X. The points y ∈ Y for which u(y) = 0 obviously form a closed set, and hence if this contains f (X), it also contains the closure f (X). Repeating the same arguments backwards, we see that f ∗(u) = 0 if and only if u vanishes on f (X), or equivalently, 32 1 Basic Notions u ∈ af (X). It follows in particular that the kernel of f ∗ is zero if and only if f (X) = Y, that is, f (X) is dense in Y. This is certainly the case if f (X) = Y, but cases with f (X) = Y but f
(X) = Y are possible (see Example 1.13). In what follows we will be concerned mainly with algebraic varieties in projective space. But closed subsets of afﬁne space have a geometry with a speciﬁc ﬂavour, which is often quite nontrivial. As an example we give the following theorem due to Abhyankar and Moh: Theorem A curve X ⊂ A2 is isomorphic to A1 if and only if there exists an automorphism of A2 that takes X to a line. (Here an automorphism is an isomorphism from A2 to itself.) The group Aut A2 of automorphisms of the plane is an extremely interesting object. Some examples of automorphisms are very simply to construct: the afﬁne linear maps, and maps of the form x = αx, y = βy + f (x), (1.20) where α, β = 0 are constants, and f a polynomial. It is known that the whole group Aut A2 is generated by these automorphisms. Moreover, the expression of an element g ∈ Aut A2 as a word in afﬁne linear maps and maps of the form (1.20) is almost unique: the only relations in Aut A2 between maps of these two classes are those expressing the fact that the two classes have a subset in common, namely maps of the form (1.20) with f a linear polynomial. In the language of abstract group theory, Aut A2 is the free product (or amalgamation) of two subgroups, the maps of the form (1.20) and the afﬁne maps, over their common subgroup (see Kurosh [53, Section 35, Chapter IX, Vol. II and Ex. 10]). A famous unsolved problem related to automorphisms of A2 is the Jacobian con- jecture. This asserts that, if the ground ﬁeld k has characteristic 0, a map given by x = f (x, y), y = g(x, y) with f, g ∈ k[x, y] is an automorphism of A2 if and only if the Jacobian determinant ∂(f,g) ∂(x,y) is a nonzero constant. At present this conjecture is proved when the degrees of f and g are not too
large (the order of 100). There is a similar conjecture for the n-dimensional afﬁne space An. 2.4 Exercises to Section 2 1 The set X ⊂ A2 is deﬁned by the equation f : x2 + y2 = 1 and g : x = 1. Find the ideal AX. Is it true that AX = (f, g)? 2 Closed Subsets of Afﬁne Space 33 2 Let X ⊂ A2 be the algebraic plane curve deﬁned by y2 = x3. Prove that an element of k[X] can be written uniquely in the form P (x) + Q(x)y with P (x), Q(x) polynomials. 3 Let X be the curve of Exercise 2 and f (t) = (t 2, t 3) the regular map A1 → X. Prove that f is not an isomorphism. [Hint: Try to construct the inverse of f as a regular map, using the result of Exercise 2.] 4 Let X be the curve deﬁned by the equation y2 = x2 + x3 and f : A1 → X the map deﬁned by f (t) = (t 2 − 1, t (t 2 − 1)). Prove that the corresponding homomorphism f ∗ maps k[X] isomorphically to the subring of the polynomial ring k[t] consisting of polynomials g(t) such that g(1) = g(−1). (Assume that char k = 2.) 5 Prove that the hyperbola deﬁned by xy = 1 and the line A1 are not isomorphic. 6 Consider the regular map f : A2 → A2 deﬁned by f (x, y) = (x, xy). Find the image f (A2); is it open in A2? Is it dense? Is it closed? 7 The same question as in Exercise 6 for the map f : A3 → A3 deﬁned by f (x, y, z) = (x, xy, xyz). 8 An isomorphism f : X → X of a closed set X to itself is called an automorphism. Prove that all automorphisms of the line A1 are of the form f
(x) = ax + b with a = 0. 9 Prove that the map f (x, y) = (αx, βy + P (x)) is an automorphism of A2, where α, β ∈ k are nonzero elements, and P (x) is a polynomial. Prove that maps of this type form a group B. 10 Prove that if f (x1,..., xn) = (P1(x1,..., xn),..., Pn(x1,..., xn)) is an automorphism of An then the Jacobian J (f ) = det \| ∂Pi \| ∈ k. Prove that f → J (f ) is ∂xj a homomorphism from the group of automorphisms of An into the multiplicative group of nonzero elements of k. 11 Suppose that X consists of two points. Prove that the coordinate ring k[X] is isomorphic to the direct sum of two copies of k. 12 Let f : X → Y be a regular map. The subset Γf ⊂ X × Y consisting of all points of the form (x, f (x)) is called the graph of f. Prove that (a) Γf ⊂ X × Y is a closed subset, and (b) Γf is isomorphic to X. 13 The map pY : X × Y → Y deﬁned by pY (x, y) = y is called the projection to Y or the second projection. Prove that if Z ⊂ X and f : X → Y is a regular map then 34 1 Basic Notions f (Z) = pY ((Z × Y ) ∩ Γf ), where Γf is the graph of f and Z × Y ⊂ X × Y is the subset of (z, y) with z ∈ Z. 14 Prove that for any regular map f : X → Y there exists a regular map g : X → X × Y that is an isomorphism of X with a closed subset of X × Y and such that f = pY ◦ g. In other words, any map is the composite of an embedding and a projection. 15 Prove that if X = Uα is any covering of a closed set X by open subsets Uα then
there exists a ﬁnite number Uα1,..., Uαr of the Uα such that X = Uα1 ∪ · · · ∪ Uαr. 16 Prove that the Frobenius map ϕ (Example 1.16) is a one-to-one correspondence. Is it an isomorphism, for example if X = A1? 17 Find the zeta function ZX(t) for X = An. 18 Determine ZX(t) for X a nonsingular conic in A2. 3 Rational Functions 3.1 Irreducible Algebraic Subsets In Section 1.1 we introduced the notion of an irreducible algebraic curve in the plane. Here we formulate the analogous notion in general. Deﬁnition A closed algebraic set X is reducible if there exist proper closed subsets X1, X2 X such that X = X1 ∪ X2. Otherwise X is irreducible. Theorem 1.4 Any closed set X is a ﬁnite union of irreducible closed sets. Proof Suppose that the theorem fails for a set X. Then X is reducible, X = X1 ∪ 1, and the theorem must fail either for X1 or for X X 1. If X1, then it is reducible, and again it is made up of closed sets one of which is reducible. In this way we construct an inﬁnite strictly decreasing chain of closed subsets X X1 X2 · · ·. We prove that there cannot be such a chain. Indeed, the ideals corresponding to the · · ·. But such an inﬁnite Xi would form an increasing chain AX AX1 strictly increasing chain cannot exist, since every ideal of the polynomial ring has a ﬁnite basis, and hence an increasing chain of ideals terminates. The theorem is proved. AX2 If X = Xi is an expression of X as a ﬁnite union of irreducible closed sets, and if Xi ⊂ Xj for some i = j then we can delete Xi from the expression. RepeatXi in which Xi ⊂ Xj ing this several times, we arrive at a representation X = 3 Rational Functions 35 for all i = j. We say that such a representation is irredundant, and the Xi are the irreducible
components of X. Theorem 1.5 The irredundant representation of X as a ﬁnite union of irreducible closed sets is unique. Proof Let X = i Xi = j Yj be two irredundant representations. Then Xi = Xi ∩ X = Xi ∩ Yj = (Xi ∩ Yj ). j j Since by assumption Xi is irreducible, we have Xi ∩ Yj = Xi for some j, that is, Xi ⊂ Yj. Repeating the argument with the Xi and Yj interchanged gives Yj ⊂ Xi for some i. Hence Xi ⊂ Yj ⊂ Xi, so that by the irredundancy of the representation, i = i and Yj = Xi. The theorem is proved. We now restate the condition that a closed set X is irreducible in terms of its coordinate ring k[X]. If X = X1 ∪ X2 is reducible then since X X1 there exists a polynomial F1 that is 0 on X1 but not 0 on X, and a similar polynomial F2 for X2. Then the product F1F2 is 0 on both X1 and X2, hence on X. The corresponding regular functions f1, f2 ∈ k[X] have the property that f1, f2 = 0, but f1f2 = 0. In other words, f1 and f2 are zerodivisors in k[X]. Conversely, suppose that k[X] has zerodivisors f1, f2 = 0, with f1f2 = 0. Write X1, X2 for the closed subsets of X corresponding to the ideals (f1) and (f2) of k[X]. In other words, Xi consists of the points x ∈ X such that fi(x) = 0, for i = 1 or 2. Obviously both Xi X, since fi = 0 on X, and X = X1 ∪ X2 since f1f2 = 0 on X, so that at each point x ∈ X either f1(x) = 0 or f2(x) = 0. Therefore, a closed set X is irreducible if and only if its coordinate ring k[X] has no zerodivisors. This in turn is equivalent to AX being a prime
ideal. If a closed subset Y is contained in X then obviously so are its irreducible components. In terms of the ring k[X] the irreducibility of a closed subset Y ⊂ X is reﬂected in aY ⊂ k[X] being a prime ideal. A hypersurface X ⊂ An with equation f = 0 is irreducible if and only if the polynomial f is irreducible. Thus our terminology is compatible with that used in Section 1 in the case of plane curves. Theorem 1.6 A product of irreducible closed sets is irreducible. Proof Suppose that X and Y are irreducible, but X × Y = Z1 ∪ Z2, with Zi X × Y for i = 1, 2. For any point x ∈ X, the closed set x × Y, consisting of points (x, y) with y ∈ Y, is isomorphic to Y, and is therefore irreducible. Since x × Y = (x × Y ) ∩ Z1 (x × Y ) ∩ Z2, ∪ either x × Y ⊂ Z1 or x × Y ⊂ Z2. Consider the subset X1 ⊂ X consisting of points x ∈ X such that x × Y ⊂ Z1; we now prove that X1 is a closed set. Indeed, for 36 1 Basic Notions any point y ∈ Y, the set Xy of points x ∈ X such that x × y ∈ Z1 is closed: it is characterised by (X × y) ∩ Z1 = Xy × y, and the left-hand side is closed as an intersection of closed sets; now X1 = y∈Y Xy is closed. In the same way, the set X2 consisting of all points x ∈ X such that x × Y ⊂ Z2 is also closed. We see that X1 ∪ X2 = X, and since X is irreducible it follows from this that X1 = X or X2 = X. In the ﬁrst case X × Y = Z1, and in the second X × Y = Z2. This contradiction proves the theorem. 3.2 Rational Functions It is known that any ring without zerodivisors can be embedded into a ﬁeld
, its ﬁeld of fractions. Deﬁnition If a closed set X is irreducible then the ﬁeld of fractions of the coordinate ring k[X] is the function ﬁeld or ﬁeld of rational functions of X; it is denoted by k(X). Recalling the deﬁnition of the ﬁeld of fractions, we can say that the function ﬁeld k(X) consists of rational functions F (T )/G(T ) such that G(T ) /∈ AX, and F /G = F1/G1 if F G1 − F1G ∈ AX. This means that the ﬁeld k(X) can be constructed as follows. Consider the subring OX ⊂ k(T1,..., Tn) of rational functions f = P /Q with P, Q ∈ k[T ] and Q /∈ AX. The functions f with P ∈ AX form an ideal MX and k(X) = OX/MX. In contrast to regular functions, a rational function on a closed set X does not necessarily have well-deﬁned values at every point of X; for example, the function 1/x at x = 0 or x/y at (0, 0). We now ﬁnd out when this is possible. Deﬁnition A rational function ϕ ∈ k(X) is regular at x ∈ X if it can be written in the form ϕ = f/g with f, g ∈ k[X] and g(x) = 0. In this case we say that the element f (x)/g(x) ∈ k is the value of ϕ at x, and denote it by ϕ(x). Theorem 1.7 A rational function ϕ that is regular at all points of a closed subset X is a regular function on X. Proof Suppose ϕ ∈ k(X) is regular at every point x ∈ X. This means that for every x ∈ X there exists fx, gx ∈ k[X] with gx(x) = 0 such that ϕ = fx/gx. Consider the ideal a generated by all the functions gx for x ∈ X. This has a ﬁ
nite basis, so that there are a ﬁnite number of points x1,..., xN such that a = (gx1,..., gxN ). The functions gxi do not have a common zero x ∈ X, since then all functions in a would vanish at x, but gx(x) = 0. From the analogue of the Nullstellensatz it follows that a = (1), and hence there exist functions u1,..., uN ∈ k[X] such that = 1. Multiplying both sides of this equality by ϕ and using the fact N i=1 uigxi 3 Rational Functions that ϕ = fxi /gxi, we get that ϕ = proved. N i=1 uifxi, that is, ϕ ∈ k[X]. The theorem is 37 If ϕ is a rational function on a closed set X, the set of points at which ϕ is regular is nonempty and open. The ﬁrst assertion follows since ϕ can be written ϕ = f/g with f, g ∈ k[X] and g = 0; hence g(x) = 0 for some x ∈ X, and obviously ϕ is regular at this point. To prove the second assertion, consider all possible representations ϕ = fi/gi. For any regular function gi the set Yi ⊂ X of points x ∈ X for which gi(x) = 0 is obviously closed, and hence Ui = X \ Yi is open. The set U of points at which ϕ is regular is by deﬁnition U = Ui, and is therefore open. This open set is called the domain of deﬁnition of ϕ. For any ﬁnite system ϕ1,..., ϕm of rational functions, the set of points x ∈ X at which they are all regular is again open and nonempty. The ﬁrst assertion follows since the intersection of a ﬁnite number of open sets is open, and the second from the following useful proposition: the intersection of a ﬁnite number of nonempty open sets of an irreducible closed set is nonempty. Indeed, let Ui = X \ Yi for i = 1,..., m be such that Ui
= ∅. Then Yi = X; but the Yi are closed sets, and this contradicts the irreducibilYi = X and ity of X. Thus for any ﬁnite set of rational functions, there is some nonempty open set on which they are all deﬁned and can be compared. This remark is useful because a rational function ϕ ∈ k(X) is uniquely determined if it is speciﬁed on some nonempty open subset U ⊂ X. Indeed, if ϕ(x) = 0 for all x ∈ U and ϕ = 0 on X then any expression ϕ = f/g with f, g ∈ k[X] gives a representation of X as a union X = X1 ∪ X2 of two closed sets, where X1 = X − U and X2 is deﬁned by f = 0. This contradicts the irreducibility of X. 3.3 Rational Maps Let X ⊂ An be an irreducible closed set. A rational map ϕ : X → Am is a map given by an arbitrary m-tuple of rational functions ϕ1,..., ϕm ∈ k(X). Thus a rational map ϕ is not a map deﬁned on the whole set X to the set Am, but it clearly deﬁnes a map of some nonempty open set U ⊂ X to Am. Working with functions and maps that are not deﬁned at all points is an essential difference between algebraic geometry and other branches of geometry, for example, topology. We now deﬁne the notion of rational map ϕ : X → Y to a closed subset Y ⊂ Am. Deﬁnition A rational map ϕ : X → Y ⊂ Am is an m-tuple of rational functions ϕ1,..., ϕm ∈ k(X) such that, for all points x ∈ X at which all the ϕi are regular, ϕ(x) = (ϕ1(x),..., ϕm(x)) ∈ Y ; we say that ϕ is regular at such a point x, and ϕ(x) ∈ Y is the image of x. The image of X under a rational map ϕ is the set of
points ϕ(X) = ϕ(x) \| x ∈ X and ϕ is regular at x. 38 1 Basic Notions As we proved at the end of Section 3.2, there exists a nonempty open set U ⊂ X on which all the rational functions ϕi are deﬁned, hence also the rational map ϕ = (ϕ1,..., ϕm). Thus we can view rational maps as maps deﬁned on open subsets; but we have to bear in mind that different maps may have different domains of deﬁnition. The same of course also applies to rational functions. To check that rational functions ϕ1,..., ϕm ∈ k(X) deﬁne a rational map ϕ : X → Y we need to check that ϕ1,..., ϕm, as elements of k(X), satisfy all the equations of Y. Indeed, if this property holds then for any polynomial u(T1,..., Tm) ∈ AY the function u(ϕ1,..., ϕm) = 0 on X. Then at each point x at which all the ϕi are regular, we have u(ϕ1(x),..., ϕm(x)) = 0 for all u ∈ AY, that is, (ϕ1(x),..., ϕm(x)) ∈ Y. Conversely, if ϕ : X → Y is a rational map, then for every u ∈ AY the function u(ϕ1,..., ϕm) ∈ k(X) vanishes on some nonempty open set U ⊂ X, and so is 0 on the whole of X. It follows from this that u(ϕ1,..., ϕm) = 0 in k(X). We now study how rational maps act on rational functions on a closed set. Let ϕ : X → Y be a rational map and assume that ϕ(X) is dense in Y. Consider ϕ as a map U → ϕ(X) ⊂ Y, where U is the domain of deﬁnition of ϕ, and construct the map ϕ∗ on functions corresponding to it. For any function
f ∈ k[Y ] the function ϕ∗(f ) is a rational function on X. Indeed, if Y ⊂ Am, and f is given by a polynomial u(T1,..., Tm), then ϕ∗(f ) is given by the rational function u(ϕ1,..., ϕm). Thus we have a map ϕ∗ : k[Y ] → k(X) which is obviously a ring homomorphism of the ring k[Y ] to the ﬁeld k(X). This homomorphism is even an isomorphic inclusion k[Y ] → k(X). Indeed, if ϕ∗(u) = 0 for u ∈ k[Y ] then u = 0 on ϕ(X). But if u = 0 on Y then the equality u = 0 deﬁnes a closed subset V (u) Y. Then ϕ(X) ⊂ V (u), but this contradicts the assumption that ϕ(X) is dense in Y. The inclusion ϕ∗ : k[Y ] → k(X) can be extended in an obvious way to an isomorphic inclusion of the ﬁeld of fractions k(Y ) into k(X). Thus if ϕ(X) is dense in Y, the rational map ϕ deﬁnes an isomorphic inclusion ϕ∗ : k(Y ) → k(X). Given two rational maps ϕ : X → Y and ψ : Y → Z such that ϕ(X) is dense in Y then it is easy to see that we can deﬁne a composite ψ ◦ ϕ : X → Z; if in addition ψ(Y ) is dense in Z then so is (ψ ◦ ϕ)(X). Then the inclusions of ﬁelds satisfy the relation (ψ ◦ ϕ)∗ = ϕ∗ ◦ ψ ∗. Deﬁnition A rational map ϕ : X → Y is birational or is a birational equivalence if ϕ has an inverse rational map ψ : Y → X, that is, ϕ(X) is dense in Y and ψ(Y ) in X, and ψ ◦ ϕ = 1, ϕ ◦ ψ =
1 (where deﬁned). In this case we say that X and Y are birational or birationally equivalent. Obviously if ϕ : X → Y is a birational map then the inclusion of ﬁelds ϕ∗ : k(Y ) → k(X) is an isomorphism. It is easy to see that the converse is also true (for algebraic plane curves this was done in Section 1.4). Thus closed sets X and Y are birational if and only if the ﬁelds k(X) and k(Y ) are isomorphic over k. Examples In Section 1 we treated a series of examples of birational maps between algebraic plane curves. Isomorphic closed sets are obviously birational. The regular maps in Examples 1.18–1.19, although not isomorphisms, are birational maps. 3 Rational Functions 39 A closed set that is birational to an afﬁne space An is said to be rational. Rational algebraic curves were discussed in Section 1. We now give some other examples of rational closed sets. Example 1.22 An irreducible quadric X ⊂ An deﬁned by a quadratic equation F (T1,..., Tn) = 0 is rational. The proof given in Section 1.2 for the case n = 2 works in general. The corresponding map can once again be interpreted as the projection of X from some point x ∈ X to a hyperplane L ⊂ An not passing through x (stereographic projection). We need only choose x so that it is not a vertex of X, that is, so that ∂F /∂Ti(x) = 0 for at least one value of i = 1,..., n. Example 1.23 Consider the hypersurface X ⊂ A3 deﬁned by the 3rd degree equation x3 + y3 + z3 = 1. We suppose that the characteristic of the ground ﬁeld k is different from 3. The surface X contains several lines, for example the two skew lines L1 and L2 deﬁned by L1 : x + y = 0, z = 1, and L2 : x + εy = 0, z = ε, where ε = 1 is a cube root of 1. We give a geometric description of a rational map
of X to the plane, and leave the reader to write out the formulas, and also to check that it is birational. Choose some plane E ⊂ A3 not containing L1 or L2. For x ∈ X \ (L1 ∪ L2), it is easy to verify that there is a unique line L passing through x and intersecting L1 and L2. Write f (x) for the point of intersection L ∩ E; then x → f (x) is the required rational map X → E. This argument obviously applies to any cubic surface in A3 containing two skew lines. In algebraic geometry we work with two different equivalence relations between closed sets, isomorphism and birational equivalence. Birational equivalence is clearly a coarser equivalence relation than isomorphism; in other words, two closed sets can be birational without being isomorphic. Thus it often turns out that the classiﬁcation of closed sets up to birational equivalence is simpler and more transparent than the classiﬁcation up to isomorphism. Since it is deﬁned at every point, isomorphism is closer to geometric notions such as homeomorphism and diffeomorphism, and so more convenient. Understanding the relation between these two equivalence relations is an important problem; the question is to understand how much coarser birational equivalence is compared to isomorphism, or in other words, how many closed sets are distinct from the point of view of isomorphism but the same from that of birational equivalence. This problem will reappear frequently later in this book. We conclude this section by proving one result that illustrates the notion of bira- tional equivalence. Theorem 1.8 Any irreducible closed set X is birational to a hypersurface of some afﬁne space Am. 40 1 Basic Notions Proof k(X) is generated over k by a ﬁnite number of elements, for example the coordinates t1,..., tn in An, viewed as functions on X. Suppose that d is the maximal number of the ti that are algebraically independent over k. According to Proposition A.7, the ﬁeld k(X) can be written in the form k(z1,..., zd+1), where z1,..., zd are algebraically independent over k and f
(z1,..., zd+1) = 0, (1.21) with f irreducible over k and f = 0. The function ﬁeld k(Y ) of the closed set Y deﬁned by (1.21) is obviously isomorphic to k(X). This means that X and Y are birational. The theorem is proved. Td+1 Remark 1.1 According to Proposition A.7, the element zd+1 is separable over the ﬁeld k(z1,..., zd ). Hence the k(z1,..., zd ) ⊂ k(X) is a ﬁnite separable ﬁeld extension. Remark 1.2 It follows from the proof of Proposition A.7 and the primitive element theorem of Galois theory that z1,..., zd+1 can be chosen as linear combinations n j =1 cij xj for i = of the original coordinates x1,..., xn, that is, of the form zi = 1,..., d + 1. The map (x1,..., xn) → (z1,..., zd+1) given by these formulas is a n projection of the space An parallel to the linear subspace deﬁned by j =1 cij xj = 0 for i = 1,..., d + 1. This shows the geometric meaning of the birational map whose existence is established in Theorem 1.8. 3.4 Exercises to Section 3 1 Suppose that k is a ﬁeld of characteristic = 2. Decompose into irreducible components the closed set X ⊂ A3 deﬁned by x2 + y2 + z2 = 0, x2 − y2 − z2 + 1 = 0. 2 Prove that if X is the closed set of Exercise 4 of Section 2.4 then the elements of the ﬁeld k(X) can be expressed in a unique way in the form u(x) + v(x)y where u(x) and v(x) are arbitrary rational functions of x. 3 Prove that the maps f of Exercises 3, 4 and 6 of Section 2.4 are bir
ational. 4 Decompose into irreducible components the closed set X ⊂ A3 deﬁned by y2 = xz, z2 = y3. Prove that all its components are birational to A1. 5 Let X ⊂ An be the hypersurface deﬁned by an equation fn−1(T1,..., Tn) + fn(T1,..., Tn) = 0, where fn−1 and fn are homogeneous polynomials of degrees n − 1 and n. (A hypersurface of this form is called a monoid.) Prove that if X is irreducible then it is birational to An−1. (Compare the case of plane curves treated in Section 1.4.) 4 Quasiprojective Varieties 41 6 At what points of the circle given by x2 + y2 = 1 is the rational function (1 − y)/x regular? 7 At which points of the curve X deﬁned by y2 = x2 + x3 is the rational function t = y/x regular? Prove that y/x /∈ k[X]. 4 Quasiprojective Varieties 4.1 Closed Subsets of Projective Space Let V be a vector space of dimension n + 1 over the ﬁeld k. The set of lines (that is, 1-dimensional vector subspaces) of V is called the n-dimensional projective space, and denoted by P(V ) or Pn. If we introduce coordinates ξ0,..., ξn in V then a point ξ ∈ Pn is given by n + 1 elements (ξ0 : · · · : ξn) of the ﬁeld k, not all equal to 0; and two points (ξ0 : · · · : ξn) and (η0 : · · · : ηn) are considered to be equal in Pn if and only if there exists λ = 0 such that ηi = λξi for i = 0,..., n. Any set (ξ0 : · · · : ξn) deﬁning the point ξ is called a set of homogeneous coordinates for ξ (compare Section 1.6). We say that
a polynomial f (S) ∈ k[S0,..., Sn] vanishes at ξ ∈ Pn if f (ξ0,..., ξn) = 0 for any choice of the coordinates (ξ0,..., ξn) of ξ. Obviously, then also f (λξ0,..., λξn) = 0 for all λ ∈ k with λ = 0. Write f in the form f = f0 + f1 + · · · + fr, where fi is the sum of all terms of degree i in f. Then f (λξ0,..., λξn) = f0(ξ0,..., ξn) + λf1(ξ0,..., ξn) + · · · + λr fr (ξ0,..., ξn). Since k is an inﬁnite ﬁeld, the equality f (λξ0,..., λξn) = 0 for all λ ∈ k with λ = 0 implies that fi(λξ0,..., λξn) = 0. Thus if f vanishes at a point ξ then all of its homogeneous components fi also vanish at ξ. Deﬁnition X ⊂ Pn is a closed subset if it consists of all points at which a ﬁnite number of polynomials with coefﬁcients in k vanish. A closed subset deﬁned by one homogeneous equation F = 0 is called a hypersurface, as in the afﬁne case. The degree of the polynomial is the degree of the hypersurface. A hypersurface of degree 2 is called a quadric. The set of all polynomials f ∈ k[S0,..., Sn] that vanish at all points x ∈ X forms an ideal of k[S], called the ideal of the closed set X, and denoted by AX. By what we said above, the ideal AX has the property that whenever it contains an element f it also contains all the homogeneous components of f. An ideal with this property is said to be
homogeneous or graded. Thus the ideal of a closed set X of projective space is homogeneous. It follows from this that it has a basis consisting of homogeneous polynomials: we need only start from any basis and take the system of 42 1 Basic Notions homogeneous components of polynomials of the basis. In particular, any closed set can be deﬁned by a system of homogeneous equations. Thus to each closed subset X ⊂ Pn there is a corresponding homogeneous ideal AX ⊂ k[S0,..., Sn]. Conversely, any homogeneous ideal A ⊂ k[S] deﬁnes a closed subset X ⊂ Pn. That is, if F1,..., Fm is a homogeneous basis of A then X is deﬁned by the system of equation F1 = · · · = Fm = 0. If this system of equations has no other solutions in the vector space V other than 0 then it is natural to take X to be the empty set. Examples of Closed Subsets of Projective Space Example 1.24 (The Grassmannian) The projective space P(V ) parametrises the 1dimensional vector subspaces L1 ⊂ V of a vector space V. The Grassmannian or Grassmann variety Grass(r, V ) plays the same role for r-dimensional vector subr V of V, and send spaces Lr ⊂ V. To deﬁne this, consider the rth exterior power r V. On a basis f1,..., fr of a vector subspace L into the element f1 ∧ · · · ∧ fr ∈ passing to another basis of the same vector subspace this element is multiplied by a nonzero element α ∈ k, the determinant of the matrix of the coordinate change, r V ) is uniquely deand hence the corresponding point of the projective space P( termined by the subspace L. Write P (L) for this point. It is easy to see that it determines the subspace L uniquely. If {ei} is a basis of V then {ei1 } is ∧ · · · ∧ eir ). The homogeneous a basis of coordinates pi1...ir of P (L) are called the Plücker coordinates of L. r V and P (L) = ∧ · · ·
∧ eir pi1...ir (ei1 i1<···<ir Except for the trivial cases of subspaces having dimension or codimension 1, not r V ) is of the form P (L), or in other words, not every element every point P ∈ P( r V is of the form f1 ∧ · · · ∧ fr with fi ∈ V. The necessary and sufﬁcient x ∈ condition for this to hold uses the notion of convolution. Let u ∈ V ∗ be a vector of 1 V = V the convolution u x is an element of the dual vector space. For x ∈ 0 V = k we set k, and is just the scalar product (u, x) or the value u(x). For x ∈ r V the convolution u x = 0 can be extended in a unique u x = 0. For any x ∈ way from x ∈ 1 V if we require the property u (x ∧ y) = (u x) ∧ y + (−1)a x ∧ (u y) for x ∈ a V. (1.22) r V ⊂ r−1 V. The element u x for u ∈ V ∗ and x ∈ Here u the convolution of u and x. Finally, for u1,..., us ∈ V ∗ the element u1 (us x) · · · ) depends only on x and y = u1 ∧ · · · ∧ us ∈ y x. Here y x ∈ r−s V if r ≥ s and y x = 0 if r < s. r V is called · · · (u2 s V ∗, and is denoted by r V to be of the form x = f1 ∧ · · · ∧ Necessary and sufﬁcient conditions for x ∈ fr are given by (y x) ∧ x = 0 for all y ∈ r−1 V ∗. (1.23) 4 Quasiprojective Varieties 43 It is obviously enough to check the conditions (1.23) for y = ui1 ∧ · · · ∧ uir−1, where {ui} is a basis of V ∗; in particular, if we take {ui} to be the basis dual to the basis {ei
} of V then (1.23) can be written in coordinates. They take the form r+1 t=1 (−1)t pi1...ir−1jt pj1...jt...jr+1 = 0 (1.24) for all sequences i1,..., ir−1 and j1,..., jr+1. The variety deﬁned in P( r V ) by the relations (1.23) or (1.24) is called the Grassmannian, and denoted by Grass(r, V ) or Grass(r, n) where n = dim V. We need a method of reconstructing a vector subspace L explicitly from its Plücker coordinates pi1...ir satisfying (1.24). Suppose for example that p1...r = 0. If p = (pi1...ir ) = P(L) then L has a basis of the form fi = ei + k>r aikek for i = 1,..., r. It follows easily from this that p1...i...rk (−1)kp1...i...rk, where we have set p1...r = 1 for convenience. = (−1)kaik, from which we get aik = Thus the open afﬁne sets pi1...ir = 0 of Grass(r, V ) are all isomorphic to the afﬁne space Ar(n−r) with coordinates aik (for i = 1,..., r and k = r + 1,..., n). We can see, for example, that in the open set p1...r = 0 (1.24) can be solved explicitly with the coordinates p1...r = 0 and p1...i...rk as free parameters. That is, if m ≥ 2 of the subscripts i1,..., ir are >r then pi1...ir = F (..., p1...i...rk,... ) (p1...r )m, where F is a form of degree m in p1...r = 0 and p1...i...rk with i ≤ r and k > r. A detailed treatment of Grassmannians is contained, for example, in the survey article Kleiman and Laksov [47]. The ﬁrst nontrivial case of
this theory is when r = 2. Then by (1.22) (u x) ∧ x = 1 2 u (x ∧ x) for u ∈ V ∗ and x ∈ 2 V. Hence (1.23) reduces to u (x ∧ x) = 0 for all u ∈ V ∗, that is, simply x ∧ x = 0. (1.25) Finally, when n = 4 we have dim tion in the Plücker coordinates p12, p13, p14, p23, p24, p34: 4 V = 1, so that (1.25) reduces to a single equa- p12p34 − p13p24 + p14p23 = 0. (1.26) Planes L ⊂ V in a 4-dimensional vector space V correspond to lines ⊂ P(V ) in projective 3-space. In this case, coordinates in V are denoted by x0, x1, x2, 44 1 Basic Notions x3 and the Plücker coordinates p01, p02, p03, p12, p13, p23, and (1.26) takes the form This is a quadric in projective 5-space P( 2 V ). p01p23 − p02p13 + p03p12 = 0. (1.27) Example 1.25 (The variety of associative algebras) Let A be an associative algebra over a ﬁeld k of rank n. Then after a choice of basis, A is determined by its multiplication table eiej = cl ij el with structure constants cl ij takes the form ∈ k. The associative condition for multiplication in A l ij cm cl lk = l il cl cm j k for i, j, k, m = 1,..., n. (1.28) this is again a system of quadratic equation in the structure constants cl ij. Multiplying all the basis elements ei by a nonzero element α−1 ∈ k has the effect of multiplying all the cl ij by α. Thus if we discard the algebra with zero multiplication, all algebras are described by points of the closed set in the projective space Pn3−1 deﬁned by (1.28). To be more precise, points of this set correspond to
associative multiplication laws written out in terms of a chosen basis e1,..., en. The change to a different basis is given by a nondegenerate n × n matrix. Thus the set of associative algebras of rank n over a ﬁeld k, up to isomorphism, is parametrised by the quotient of the set deﬁned by (1.28) by the group of nondegenerate n × n matrixes. The extent to which this type of quotient can be identiﬁed with an algebraic variety is an extremely delicate question. n+1 2 Example 1.26 (Determinantal varieties) Quadratic forms in n variables form a = (1/2)n(n + 1). Quadrics in an (n − 1)vector space V of dimension dimensional projective space are parametrised by points of the projective space P(V ). Among these, the degenerate quadrics are characterised by det(f ) = 0, where f is the corresponding quadratic form. This is a hypersurface X1 ⊂ P(V ). The quadrics of rank ≤n − k correspond to points of a set Xk deﬁned by setting all (n − k + 1) × (n − k + 1) minors of the matrix of f to 0. A set of this type is called a determinantal variety. Another type of determinantal variety Mk is deﬁned in the space P(V ), where V is the space of n × m matrixes, by the condition that a matrix has rank ≤k. In the case of closed subsets of afﬁne space, an ideal A ⊂ k[T ] deﬁnes the empty set only if A = (1); this is the assertion of the Nullstellensatz. For closed subsets of projective spaces this is not the case: for example, the ideal (S0,..., Sn) also 4 Quasiprojective Varieties 45 deﬁnes the empty set. Write Is for the ideal of k[S] consisting of polynomials having only terms of degree ≥s. Obviously Is also deﬁnes the empty set: it contains, for i for i = 0,..., n, which have a common zero only at
the example, the monomials Ss origin. Lemma 1.1 A homogeneous ideal A ⊂ k[S] deﬁnes the empty set if and only if it contains the ideal Is for some s > 0. Proof We have already seen that the ideal Is deﬁnes the empty set, and the same holds a fortiori for any ideal containing Is. Suppose that a homogeneous ideal A ⊂ k[S] deﬁnes the empty set. Let F1,..., Fr be a homogeneous basis of the ideal A and set deg Fi = mi. Then from the assumption, it follows that the polynomials Fi(1, T1,..., Tn) have no common root, where Tj = Sj /S0. Indeed, a common root (α1,..., αn) would give a common root (1, α1,..., αn) of F1,..., Fr. By the Nullstellensatz there must exist polynomials Gi(T1,..., Tn) such i Fi(1, T1,..., Tn)Gi(T1,..., Tn) = 1. Setting Tj = Sj /S0 in this equality that 0 we get Sl0 and multiplying through by a common denominator of the form Sl0 ∈ A. 0 In the same way, for each i = 1,..., n there exists a number li > 0 such that Sli ∈ A. i If now l = max(l0,..., ln) and s = (l − 1)(n + 1) + 1 then in any term Sa0 · · · San n of 0 degree a0 + · · · + an ≥ s we must have at least one term Si with exponent ai ≥ l ≥ li, and since Sli ∈ A, this term is contained in A. This proves that Is ⊂ A. The lemma i is proved. From now on we consider closed subsets of afﬁne and projective spaces at one and the same time. We again call these afﬁne and projective closed sets. For projective closed sets, we use the same terminology as for afﬁne sets; that is, if Y �
� X are two closed sets then we say that X \ Y is an open set in X. As before, a union of an arbitrary number of open sets, and an intersection of ﬁnitely many open sets ⊂ Pn of points ξ = (ξ0 : · · · : ξn) for which ξ0 = 0 is obis again open. The set An 0 viously open. Its points can be put in one-to-one correspondence with the points of an n-dimensional afﬁne space by setting αi = ξi/ξ0 for i = 1,..., n, and sending ξ ∈ An 0 an afﬁne piece of Pn. In the 0 to (α1,..., αn) ∈ An. Thus we call the set An i consists of points for which ξi = 0. Obviously same way, for i = 0,..., n, the set An Pn = For any projective closed set X ⊂ Pn, and any i = 0,..., n, the set Ui = X ∩ An i is open in X. It is closed as a subset of An i. Indeed, if X is given by a system of homogeneous equations F1 = · · · = Fm = 0 and deg Fj = nj then, for example, U0 is given by the system An i. i S −nj 0 Fj = Fj (1, T1,..., Tn) = 0 for j = 1,..., m, where Ti = Si/S0 for i = 1,..., n. We call Ui the afﬁne pieces of X; obviously X = 0 deﬁnes a closed projective set U called its projective completion; U is the intersection of all projective closed sets containing U. It is easy to check that the homogeneous equations of U are obtained by a process inverse Ui. A closed subset U ⊂ An 46 1 Basic Notions to that just described. If F (T1,..., Tn) is any polynomial in the ideal A of U of degree deg F = k, then the equations of U are of the form Sk 0 F (S1/S0,..., Sn/S
0). It follows from this that U = U ∩ An 0. (1.29) Up to now we have considered two classes of objects that could claim to be called algebraic varieties; afﬁne and projective closed sets. It is natural to try to introduce a uniﬁed notion of which both of these types will be particular cases. This will be done most systematically in Chapters 5–6 in connection with the notion of scheme. For the moment we introduce a more particular notion, that uniﬁes projective and afﬁne closed sets. Deﬁnition A quasiprojective variety is an open subset of a closed projective set. A closed projective set is obviously a quasiprojective variety. For afﬁne closed sets this follows from (1.29). A closed subset of a quasiprojective variety is its intersection with a closed set of projective space. Open set and neighbourhood of a point are deﬁned similarly. The notion of irreducible variety and the theorem on decomposing a variety as a union of irreducible components carries over word-for-word from the case of afﬁne sets. From now on we use subvariety Y of a quasiprojective variety X ⊂ Pn to mean any subset Y ⊂ X which is itself a quasiprojective variety in Pn. This is obviously equivalent to saying that Y = Z \ Z1 with Z and Z1 ⊂ X closed subsets. 4.2 Regular Functions We proceed to considering functions on quasiprojective varieties, and start with the projective space Pn itself. Here we meet an important distinction between functions of homogeneous and inhomogeneous coordinates: a rational function of the homogeneous coordinates f (S0,..., Sn) = P (S0,..., Sn) Q(S0,..., Sn) (1.30) cannot be viewed as a function of x ∈ Pn, even when Q(x) = 0, since the value f (α0,..., αn) in general changes when all the αi are multiplied through by a common factor. However, when f is a homogeneous function of degree 0, that is, when P and Q are homogeneous of the same degree
, then f can be viewed as a function of x ∈ Pn. If X ⊂ Pn is a quasiprojective variety, x ∈ X and f = P /Q is a homogeneous function of degree 0 with Q(x) = 0, then f deﬁnes a function on a neighbourhood of x in X with values in k. We say that f is regular in a neighbourhood of x, or simply at x. A function on X that is regular at all points x ∈ X is a regular function on X. All regular functions on X form a ring, that we denote by k[X]. 4 Quasiprojective Varieties 47 Let’s prove that for a closed subset X of an afﬁne space, our deﬁnition of regular function here is the same as that in Section 2.2 (after an obvious passage to inhomogeneous coordinates). For X irreducible, this is stated in Theorem 1.7. In the general case we only need to change slightly the arguments used to prove this theorem. In this proof we let f be a regular function in the afﬁne sense of Section 2.2. By assumption, each point x ∈ X has a neighbourhood Ux with qx = 0 on Ux in which f = px/qx, where px, qx are regular functions on X and qx = 0 on Ux. Hence qxf = px (1.31) on Ux. But we can assume that (1.31) holds over the whole of X. To achieve this, we multiply both px and qx by a regular function equal to 0 on X \ Ux and nonzero at x; then (1.31) holds also on X \ Ux, since both sides are 0 there. As in the proof of Theorem 1.7, we can ﬁnd points x1,..., xN ∈ X and regular functions h1,..., hN N i=1 qxi hi = 1. Multiply (1.31) for x = xi by hi and add, to get such that f = N i=1 pxi hi, that is, f is a regular function. In contrast to the case of closed afﬁne sets, the ring k[X] may consist only of constants. We
will prove later (Theorem 1.11, Corollary 1.1) that this is always the case if X is an irreducible closed projective set. This is easy to prove directly if X = Pn: indeed, if f = P /Q, with P and Q forms of the same degree, we can assume that P and Q have no common factors; then f is not regular at points x where Q(x) = 0. On the other hand, when X is only quasiprojective, k[X] may turn out to be an unexpectedly large ring. If X is an afﬁne closed set then as we have seen k[X] is ﬁnitely generated as an algebra over k. However, Rees and Nagata constructed examples of quasiprojective varieties for which k[X] is not ﬁnitely generated. This shows that k[X] is only a reasonable invariant when X is an afﬁne closed set. We pass to maps. Any map of a quasiprojective variety X to an afﬁne space An is given by n functions on X with values in k. If these functions are regular then we say the map is regular. Deﬁnition Let f : X → Y be a map between quasiprojective varieties, with Y ⊂ Pm. This map is regular if for every point x ∈ X and for some afﬁne piece Am i containing f (x) there exists a neighbourhood U x such that f (U ) ⊂ Am i and the map f : U → Am i is regular. We check that the regularity property is independent of the choice of afﬁne piece i containing f (x). If f (x) = (y0,...,1,..., ym) ∈ Am (where 1 in the ith place Am j, then yj = 0, and the means that this coordinate is discarded) is also contained in Am j are (y0/yj,..., 1/yj,...,1,... ym/yj ), with 1/yj coordinates of this point in Am i 48 1 Basic Notions in the ith place and 1 discarded from the j th. Therefore if f : U → Am i functions (f
0,...,1,..., fm), the map f to Am j is given by (f0/fj,..., 1/fj,...,1,... fm/fj ). is given by By assumption fj (x) = 0, and the subset U ⊂ U of points at which fj = 0 is open. The functions f0/fj,..., 1/fj,..., fm/fj are regular on U, and hence f : U → Am j is regular. In the same way as for afﬁne closed sets, a regular map f : X → Y deﬁnes a homomorphism f ∗ : k[Y ] → k[X]. The question of how to write down formulas deﬁning a regular map on an irreducible variety is solved in complete analogy with the case n = 2 treated in Section 1.6. Suppose for example that f (x) ∈ Am 0 is given by regular functions f1,..., fm. By deﬁnition fi = Pi/Qi where Pi, Qi are forms of the same degree in the homogeneous coordinates of x and Qi(x) = 0. Putting these fractions over a common denominator gives fi = Fi/F0, where F0,..., Fm are forms of the same degree and F0(x) = 0. In other words, f (x) = (F0(x) : · · · : Fm(x)) ∈ Pm. In this process, we must bear in mind that the representation of a regular function as a ratio of two forms is not unique. Hence two different formulas 0, and the map f : U → Am f (x) = F0(x) : · · · : Fm(x) and g(x) = G0(x) : · · · : Gm(x) (1.32) may deﬁne the same map; this happens if and only if FiGj = Fj Gi on X for 0 ≤ i, j ≤ m. (1.33) This brings us to a second form of the deﬁnition of a regular map: Deﬁnition A regular map f : X → Pm of
an irreducible quasiprojective variety X to projective space Pm is given by an (m + 1)-tuple of forms (F0 : · · · : Fm) (1.34) of the same degree in the homogeneous coordinates of x ∈ Pn. We require that for every x ∈ X there exists an expression (1.34) for f such that Fi(x) = 0 for at least one i; then we write f (x) to denote the point (F0(x) : · · · : Fm(x)). Two maps (1.32) are considered equal if (1.33) holds. Now we have a deﬁnition of regular maps between quasiprojective varieties, it is natural to deﬁne an isomorphism to be a regular map having an inverse regular map. A quasiprojective variety X isomorphic to a closed subset of an afﬁne space will be called an afﬁne variety. It can happen that X is given as a subset X ⊂ An, but is not closed in An. For example, the set X = A1 \ 0 is not closed in A1, although it is quasiprojective, and is isomorphic to the hyperbola xy = 1 (Example 1.13), which is a closed set of A2. Thus the notion of a closed afﬁne set is not invariant under isomorphism, while that of afﬁne variety is invariant by deﬁnition. 4 Quasiprojective Varieties 49 In the same way, a quasiprojective variety isomorphic to a closed projective set will be called a projective variety. We will prove in Theorem 1.10 that if X ⊂ Pn is a projective variety then it is closed in Pn, so that the notions of closed projective set and projective variety coincide and are both invariant under isomorphism. There are quasiprojective varieties that are neither afﬁne nor projective (see Ex- ercise 5 of Section 4.5 and Exercises 4–6 of Section 5.5). In what follows, we will meet some properties of varieties X that need only be veriﬁed for some neighbourhood U of any point x
∈ X. In other words, if X = Uα, with Uα any open sets, then it is enough to verify the property for each of the Uα. We say that properties of this type are local properties. We give some example of local properties. Lemma 1.2 The property that a subset Y ⊂ X is closed in a quasiprojective variety X is a local property. Proof The assertion means that if X = Uα with open sets Uα, and Y ∩ Uα is closed in each Uα then Y is closed in X. By deﬁnition of open sets, Uα = X \ Zα where the Zα are closed, and by deﬁnition of closed sets, Uα ∩ Y = Uα ∩ Tα where the Tα ⊂ X are closed. We check that Y = (Zα ∪ Tα), from which it follows of course that Y is closed. If y ∈ Y and y ∈ Uα then y ∈ Uα ∩ Y ⊂ Tα, and if y /∈ Uα then y ∈ X \ Uα = Zα, so that y ∈ Zα ∪ Tα for every α. Conversely, suppose that x ∈ Zα ∪ Tα for every α. Since X = Uα it follows that x ∈ Uβ for some β. Then x /∈ Zβ, and hence x ∈ Tβ, so that x ∈ Tβ ∩ Uβ ⊂ Y. The lemma is proved. In studying local properties we can restrict ourselves to afﬁne varieties in view of the following result. Lemma 1.3 Every point x ∈ X has a neighbourhood isomorphic to an afﬁne variety. 0, and by deﬁnition of a quasiprojective variety X ∩ An 0 Proof By assumption X ⊂ Pn. If x ∈ An 0 (that is, if the coordinate u0 of x is nonzero) then x ∈ X ∩ An = Y \ Y1 where Y and Y1 ⊂ Y are closed subsets of An 0. Since x ∈ Y \ Y1, there exists a polynomial F of the coordinates of An 0 such that F = 0 on Y1 and F (x) = 0. Write V
(F ) for the set of points of Y where F = 0. Obviously D(F ) = Y \ V (F ) is a neighbourhood of x. We prove that this neighbourhood is isomorphic to an afﬁne variety. Suppose that G1 = · · · = Gm = 0 are the equations of Y in An 0. Deﬁne a variety Z ⊂ An+1 by the equations G1(T1,..., Tn) = · · · = Gm(T1,..., Tn) = 0, F (T1,..., Tn) · Tn+1 = 1. (1.35) The map ϕ : (x1,..., xn+1) → (x1,..., xn) obviously deﬁnes a regular map Z → D(F ) and ψ : (x1,..., xn) → (x1,..., xn, F (x1,..., xn)−1) a regular map D(F ) → Z inverse to ϕ. This proves the lemma. 50 1 Basic Notions If Y = A1, F = T then the isomorphism just constructed is the map considered in Example 1.13. Deﬁnition An open set D(f ) = X \ V (f ) consisting of the points of an afﬁne variety X such that f (x) = 0 is called a principal open set. The signiﬁcance of these sets is that they are afﬁne, as we have seen, and the ring k[D(f )] of regular function on them can be easily determined. Namely, by construction f = 0 on D(f ), so that f −1 ∈ k[D(f )], and Theorem 1.7 together with (1.35) shows that k[D(f )] = k[X][f −1]. Lemmas 1.2–1.3 show for example that closed subsets map to closed subsets under isomorphisms. We prove in addition that the inverse image f −1(Z) under any regular map f : X → Y of any closed subset Z ⊂ Y is closed in X. By deﬁnition of a regular map f
colonX → Y, for any point x ∈ X there are neighbourhoods U of x in X and V of f (x) in Y such that f (U ) ⊂ V ⊂ Am and the map f : U → V is regular. By Lemma 1.3 we can assume that U is an afﬁne variety. By Lemma 1.2, it is enough to check that f −1(Z) ∩ U = f −1(Z ∩ V ) is closed in U. Since Z ∩ V is closed in V, it is deﬁned by equations g1 = · · · = gm = 0, where the gi are regular functions on V. But then f −1(Z ∩ V ) is deﬁned by the equations f ∗(g1) = · · · = f ∗(gm) = 0, and is hence also closed. It follows also from what we have just proved that the inverse image of an open set is again open. It is easy to check that a regular map can be deﬁned as a map f : X → Y such that the inverse image of any open set is open (that is, f is “continuous”), and for any point x ∈ X and any function ϕ regular in a neighbourhood of f (x) ∈ Y, the function f ∗(ϕ) is regular in a neighbourhood of x. 4.3 Rational Functions In discussing the deﬁnition of rational functions on quasiprojective varieties, we met a distinction of substance between the case of afﬁne varieties and the general case. Namely, we deﬁned rational functions on an afﬁne variety X as ratios of functions that are regular on the whole of X. But in the general case, as we have said, it can happen that there are no everywhere regular functions except for the constants, so that if we used the same deﬁnition there would also be no rational functions except for the constants. For this reason we deﬁne rational functions on a quasiprojective variety X ⊂ Pn to be functions deﬁned on X by homogeneous functions on Pn (as in Section 1.6 for n = 2). More precisely, consider an irreducible quasiproject
ive variety X ⊂ Pn and (by analogy with Section 3.2) write OX for the set of rational functions f = P /Q in the homogeneous coordinates S0,..., Sn such that P, Q are forms of the same degree and Q /∈ AX. As for afﬁne varieties, from the fact that X is irreducible it follows that OX is a ring. Write MX for the set of functions f ∈ OX with P ∈ AX. Obviously the quotient ring OX/MX is a ﬁeld, called the function ﬁeld of X, and denoted by k(X). 4 Quasiprojective Varieties 51 If U is an open subset of an irreducible quasiprojective variety X then, since a form vanishes on X if and only if it vanishes on U, we have k(X) = k(U ). In particular, k(X) = k(X), where X is the projective closure of X in Pn. Thus in discussing function ﬁelds we can restrict to afﬁne or projective varieties if we want to. It is easy to check that if X is an afﬁne variety then the deﬁnition just given coincides with that given in Section 3.2. Indeed, dividing the numerator and denominator of a rational function f = P /Q with deg P = deg Q = m by Sm 0, we can write it as a rational function in Ti = Si/S0 for i = 1,..., n. By doing this, we establish an isomorphism of the ﬁeld of homogeneous rational functions of degree 0 in S0,..., Sn with the ﬁeld k(T1,..., Tn). An obvious veriﬁcation shows that the subring and ideal of k(T1,..., Tn) denoted in Section 3.2 by OX and MX correspond to the objects denoted here by the same letters. In Section 4.2 we have already used rational functions on Pn to deﬁne regular functions. As there, we say that f ∈ k(X) is regular at a point x ∈ X if it can be written in the form f = F
/G, with F and G homogeneous of the same degree and G(x) = 0. Then f (x) = F (x)/G(x) is the value of f at x. As in the case of afﬁne varieties, the set of points at which a given rational function f is regular is a nonempty open set U of X, called the domain of deﬁnition of f. Obviously a rational function can also be deﬁned as a function regular on some open set U ⊂ X. A rational map f : X → Pm is deﬁned (as in the second deﬁnition of regular map in Section 4.2) by giving m + 1 forms (F0 : · · · : Fm) of the same degree in the n + 1 homogeneous coordinates of Pn containing X. Here at least one of the forms must not vanish on X. Two maps (F0 : · · · : Fm) and (G0 : · · · : Gm) are equal if FiGj = Fj Gi on X for all i, j. If we divide through all the forms Fi by one of them (nonzero on X), we can deﬁne a rational map by m + 1 rational functions on X, with the same notion of equality of maps. If a rational map f can be deﬁned by functions (f0 : · · · : fm) such that all the fi are regular at x ∈ X and not all zero at x, then f is regular at x. It then deﬁnes a regular map of some neighbourhood of the point x to Pm. The set of points at which a rational map is regular is open. Hence we can also deﬁne a rational map to be a regular map of some open set U ⊂ X. If Y ⊂ Pm is a quasiprojective variety and f : X → Pm a rational map, we say that f maps X to Y if there exists an open set U ⊂ X on which f is regular and f (U ) ⊂ Y. The union U of all such open sets is called the domain of deﬁnition of f, and f (U ) ⊂ Y the image of X in Y. As in the case of afﬁne
varieties, if the image of a rational map f : X → Y is dense in Y then f deﬁnes an inclusion of ﬁelds f ∗ : k(Y ) → k(X). If a rational map f : X → Y has an inverse rational map then f is birational or is a birational equivalence, and X and Y are birational. In this case the inclusion of ﬁelds f ∗ : k(Y ) → k(X) is an isomorphism. We can now clarify the relation between the notions of isomorphism and bira- tional equivalence. Proposition 1.1 Two irreducible varieties X and Y are birational if and only if they contain isomorphic open subsets U ⊂ X and V ⊂ Y. 52 1 Basic Notions Proof Indeed, suppose that f : X → Y is birational, and let g = f −1 : Y → X be the inverse rational map. Write U1 ⊂ X and V1 ⊂ Y for the domain of deﬁnition of f and g. Then by assumption f (U1) is dense in Y, so that f −1(V1) ∩ U1 is nonempty, and as proved in Section 4.2, is open. Set U = f −1(V1) ∩ U1 and V = g−1(U1) ∩ V1. A simple check shows that f (U ) = V, g(V ) = U and fg = 1, gf = 1, that is, U and V are isomorphic. 4.4 Examples of Regular Maps Example 1.27 (Projection) Let E be a d-dimensional linear subspace of Pn deﬁned by n − d linearly independent linear equations L1 = · · · = Ln−d = 0, with Li linear forms. The projection with centre E is the rational map π(x) = (L1(x) : · · · : Ln−d (x)). This map is regular on Pn \ E, since at every point of this set one of the forms Li does not vanish. Hence if X is any closed subvariety of Pn disjoint from E, the restriction of π deﬁnes a regular map π : X → Pn−d−1
. The geometric meaning of projection is as follows: as a model of Pn−d−1 take any (n − d − 1)-dimensional linear subspace H ⊂ Pn disjoint from E. Then there is a unique (d + 1)-dimensional linear subspace E, x passing through E and any point x ∈ Pn \ E. This subspace intersects H in a unique point, which is π(x). If X intersects E, but is not contained in it, then projection from E is a rational map on X. The case d = 0, a projection from a point, has already appeared several times. Example 1.28 (The Veronese embedding) Consider all the homogeneous polynomials F of degree m in variables S0,..., Sn. These form a vector space, whose dimension is easy to compute: it is the binomial coefﬁcient. Consider the hypersurfaces of degree m in Pn. Since polynomials deﬁne the same hypersurface if and only if they are proportional, hypersurfaces correspond to points − 1. Write vi0...in for of the projective space PN of dimension N = νn,m = homogeneous coordinates of PN, where i0,..., in ≥ 0 are any nonnegative integers such that i0 + · · · + in = m. Consider the map vm : Pn → PN deﬁned by n+m m n+m m vi0...in = ui0 0 · · · uin n for i0 + · · · + in = m. (1.36) This is obviously a regular map, since the monomials on the right-hand side of i, which vanish only if all ui = 0. The (1.36) include in particular the elements um map vm is called the mth Veronese embedding of Pn, and the image vm(Pn) ⊂ PN the Veronese variety. It follows from (1.36) that the relations vi0...invj0...jn = vk0...kn vl0...ln (1.37) hold on vm(Pn) whenever i0 + j0 = k0 + l0,..., in + jn = kn + ln. Conversely, it’s

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