22:
712:
663:
614:
481:
1527:) action on a projective line, replacing "field" by "KT-field" (generalizing the inverse to a weaker kind of involution), and "PGL" by a corresponding generalization of projective linear maps.
1826:
is a rational curve that lies in no proper linear subspace; it is known that there is only one example (up to projective equivalence), given parametrically in homogeneous coordinates as
136:. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a
540:
1018:
1110:
369:
1446:
884:
264:
1956:
1262:
points. In all other respects it is no different from projective lines defined over other types of fields. In the terms of homogeneous coordinates ,
669:
620:
563:
1715:. This is the beginning of methods in algebraic geometry that are inductive on dimension. The rational maps play a role analogous to the
405:
2379:
1519:. Indeed, a generalized converse is true: a sharply 3-transitive group action is always (isomorphic to) a generalized form of a PGL
143:
There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a
1598:
86:
2471:
2108:
2068:
1949:
58:
1335:
2537:
2159:
2058:
65:
2527:
1922:
105:
1780:
may give an indeterminate result after a rational map.) This gives a picture in which the main geometric feature is
2237:
1942:
39:
1869:
72:
2384:
2305:
2295:
2232:
43:
1982:
1770:) will in fact be everywhere defined. (That is not the case if there are singularities, since for example a
54:
2567:
2202:
2098:
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2425:
2124:
2037:
1864:
1796:
2435:
2073:
1781:
1512:
1331:
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2481:
1186:
1165:
2394:
2374:
2310:
2227:
2129:
2088:
1859:
2285:
2093:
1724:
1387:
1228:
213:
32:
2078:
1807:
1548:
1471:
1168:
504:
2192:
890:
2456:
2154:
2103:
1992:
1912:
1819:
1024:
79:
287:
2532:
2404:
2063:
1716:
1568:
1560:
1395:
1126:
274:
187:
2315:
8:
2369:
2247:
2212:
2169:
2149:
1815:
1788:
1676:
1552:
727:
144:
2499:
2290:
2270:
2083:
1622:
1220:
2242:
223:
2399:
2346:
2217:
2032:
2027:
1918:
1657:
1610:
1347:
1292:
1145:
384:
208:
132:
2389:
2275:
2252:
1874:
1811:
1720:
1224:
1216:
171:
137:
2504:
2320:
2262:
2164:
1966:
1849:
1734:
is now taken to be of dimension 1, we get a picture of a typical algebraic curve
1536:
159:
127:
2187:
2012:
1997:
1974:
1792:
1207:
2556:
2519:
2300:
2280:
2207:
2002:
1934:
1836:
1571:. In general a (non-singular) curve of genus 0 is rationally equivalent over
1198:
1582:, which is itself birationally equivalent to projective line if and only if
717:
Translating this arithmetic in terms of homogeneous coordinates gives, when
2466:
2440:
2430:
2420:
2222:
2042:
1879:
1772:
1680:
1240:
1172:
167:
1695:), that is not constant. The image will omit only finitely many points of
1463:
lying in it, but the symmetries of the projective line can move the point
2341:
2179:
1854:
1516:
1315:
1161:
1132:
178:
119:
2336:
1311:
170:. This definition is a special instance of the general definition of a
707:{\displaystyle x+\infty =\infty \quad {\text{if}}\quad x\not =\infty }
2197:
1503:). This amount of specification 'uses up' the three dimensions of PGL
658:{\displaystyle x\cdot \infty =\infty \quad {\text{if}}\quad x\not =0}
609:{\displaystyle {\frac {1}{0}}=\infty ,\qquad {\frac {1}{\infty }}=0,}
21:
1898:
1594:
can be used as origin to make explicit the birational equivalence.
1176:
182:
1470:
Much more is true, in that some transformation can take any given
1362:
says that there exists a homography that will transform any point
476:{\displaystyle \left\{\in \mathbf {P} ^{1}(K)\mid x\in K\right\}.}
2509:
2494:
2489:
1358:) to emphasise the projective nature of these transformations.
1202:
1750:
is non-singular (which is no loss of generality starting with
1451:
express a one-dimensional subspace by a single non-zero point
1192:
1576:
1667:, other than a single point, has a subfield isomorphic with
1799:, the genus then depends only on the type of ramification.
1917:, Graduate Texts in Mathematics, vol. 133, Springer,
1205:. Hence the complex projective line is also known as the
140:
meet in exactly one point (there is no "parallel" case).
374:
1398:
1027:
893:
730:
672:
623:
566:
507:
408:
290:
226:
379:
The projective line may be identified with the line
1535:The projective line is a fundamental example of an
46:. Unsourced material may be challenged and removed.
1758:)), it can be shown that such a rational map from
1440:
1104:
1012:
878:
706:
657:
608:
534:
475:
363:
258:
1467:to any other, and it is in no way distinguished.
2554:
1539:. From the point of view of algebraic geometry,
1156:An example is obtained by projecting points in
126:is, roughly speaking, the extension of a usual
1964:
1153:creating a closed loop or topological circle.
1950:
1703:), and the inverse image of a typical point
277:if they differ by an overall nonzero factor
1201:results in a space that is topologically a
1193:Complex projective line: the Riemann sphere
273:that are not both zero. Two such pairs are
1957:
1943:
1515:. The computational aspect of this is the
199:An arbitrary point in the projective line
194:
1795:of the projective line. According to the
1139:. It may also be thought of as the line
106:Learn how and when to remove this message
545:This allows to extend the arithmetic on
1511:); in other words, the group action is
1120:
2555:
2380:Clifford's theorem on special divisors
1910:
1149:â; the point connects to both ends of
1938:
1530:
1234:
1227:theory, as the simplest example of a
391:may be identified with the subset of
1899:Action of PGL(2) on Projective Space
1601:of the projective line is the field
375:Line extended by a point at infinity
44:adding citations to reliable sources
15:
1563:, it is the unique such curve over
13:
2538:Vector bundles on algebraic curves
2472:Weber's theorem (Algebraic curves)
2069:Hasse's theorem on elliptic curves
2059:Counting points on elliptic curves
1914:Algebraic Geometry: A First Course
1791:, may be presented abstractly, as
1679:, this means that there will be a
1197:Adding a point at infinity to the
701:
685:
679:
636:
630:
592:
580:
508:
494:) except one, which is called the
14:
2579:
1305:
486:This subset covers all points in
158:), as the set of one-dimensional
1840:for the first interesting case.
1350:for the group, often written PGL
1189:, which distinguishes â and ââ.
1175:we can take the quotient by the
434:
217:, which take the form of a pair
20:
2160:Hurwitz's automorphisms theorem
1870:Projectively extended real line
1266:of these points have the form:
694:
688:
645:
639:
586:
31:needs additional citations for
2385:Gonality of an algebraic curve
2296:Differential of the first kind
1904:
1901:â see comment and cited paper.
1892:
1435:
1417:
1411:
1399:
1310:Quite generally, the group of
1096:
1070:
1055:
1028:
1004:
958:
952:
926:
920:
894:
870:
844:
798:
795:
789:
763:
757:
731:
526:
514:
450:
444:
426:
414:
355:
323:
317:
291:
253:
227:
1:
2528:BirkhoffâGrothendieck theorem
2238:Nagata's conjecture on curves
2109:SchoofâElkiesâAtkin algorithm
1983:Five points determine a conic
1885:
1675:). From the point of view of
1590:; geometrically such a point
1131:The projective line over the
177:The projective line over the
2099:Supersingular elliptic curve
1723:, and indeed in the case of
1637:are precisely the group PGL
1617:, in a single indeterminate
1322:acts on the projective line
1215:). It is in constant use in
7:
2306:Riemann's existence theorem
2233:Hilbert's sixteenth problem
2125:Elliptic curve cryptography
2038:Fundamental pair of periods
1865:Projective line over a ring
1843:
1727:the two concepts coincide.
1239:The projective line over a
1143:together with an idealised
1115:
387:. More precisely, the line
207:) may be represented by an
10:
2584:
2436:Moduli of algebraic curves
1810:to a projective line (see
1386:of choice of coordinates:
1124:
2518:
2480:
2449:
2413:
2362:
2355:
2329:
2261:
2178:
2142:
2117:
2051:
2020:
2011:
1973:
1787:Many curves, for example
1586:has a point defined over
1187:extended real number line
535:{\displaystyle \infty =.}
2203:CayleyâBacharach theorem
2130:Elliptic curve primality
1725:compact Riemann surfaces
1465:â = [1 : 0]
1013:{\displaystyle \cdot =,}
2462:RiemannâHurwitz formula
2426:GromovâWitten invariant
2286:Compact Riemann surface
2074:Mazur's torsion theorem
1808:birationally equivalent
1797:RiemannâHurwitz formula
1388:homogeneous coordinates
1229:compact Riemann surface
1105:{\displaystyle ^{-1}=.}
214:homogeneous coordinates
195:Homogeneous coordinates
2079:Modular elliptic curve
1442:
1298:may be represented as
1169:diametrically opposite
1106:
1014:
880:
708:
659:
610:
536:
477:
365:
364:{\displaystyle \sim .}
260:
1993:Rational normal curve
1860:Möbius transformation
1820:rational normal curve
1717:meromorphic functions
1707:will be of dimension
1490:to any other 3-tuple
1443:
1441:{\displaystyle \sim }
1107:
1015:
881:
709:
660:
611:
537:
478:
366:
261:
162:of a two-dimensional
2533:Stable vector bundle
2405:Weil reciprocity law
2395:RiemannâRoch theorem
2375:BrillâNoether theory
2311:RiemannâRoch theorem
2228:Genusâdegree formula
2089:MordellâWeil theorem
2064:Division polynomials
1911:Harris, Joe (1992),
1822:in projective space
1789:hyperelliptic curves
1569:rational equivalence
1561:algebraically closed
1513:sharply 3-transitive
1499:of distinct points (
1396:
1300:[1 : 0]
1171:points. In terms of
1137:real projective line
1127:Real projective line
1121:Real projective line
1025:
891:
728:
719:[0 : 0]
670:
621:
564:
505:
406:
288:
224:
188:Real projective line
130:by a point called a
40:improve this article
2568:Projective geometry
2356:Structure of curves
2248:Quartic plane curve
2170:Hyperelliptic curve
2150:De Franchis theorem
2094:NagellâLutz theorem
1806:is a curve that is
1677:birational geometry
1648:Any function field
1645:) discussed above.
1623:field automorphisms
1501:triple transitivity
1366:to any other point
879:{\displaystyle +=,}
150:, commonly denoted
2363:Divisors on curves
2155:Faltings's theorem
2104:Schoof's algorithm
2084:Modularity theorem
1611:rational functions
1531:As algebraic curve
1438:
1382:) is therefore an
1291:and the remaining
1235:For a finite field
1221:algebraic geometry
1211:(or sometimes the
1102:
1010:
876:
704:
655:
606:
557:) by the formulas
532:
473:
361:
256:
2550:
2549:
2546:
2545:
2457:HasseâWitt matrix
2400:Weierstrass point
2347:Smooth completion
2316:TeichmĂŒller space
2218:Cubic plane curve
2138:
2137:
2052:Arithmetic theory
2033:Elliptic integral
2028:Elliptic function
1738:presented 'over'
1658:algebraic variety
1372:point at infinity
1348:homogeneous space
1146:point at infinity
692:
643:
595:
575:
496:point at infinity
385:point at infinity
209:equivalence class
133:point at infinity
116:
115:
108:
90:
55:"Projective line"
2575:
2563:Algebraic curves
2390:Jacobian variety
2360:
2359:
2263:Riemann surfaces
2253:Real plane curve
2213:Cramer's paradox
2193:BĂ©zout's theorem
2018:
2017:
1967:algebraic curves
1959:
1952:
1945:
1936:
1935:
1929:
1927:
1908:
1902:
1896:
1875:Projective range
1812:rational variety
1721:complex analysis
1714:
1489:
1466:
1462:
1447:
1445:
1444:
1439:
1301:
1286:
1275:
1261:
1225:complex manifold
1217:complex analysis
1181:
1111:
1109:
1108:
1103:
1095:
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1082:
1081:
1066:
1065:
1053:
1052:
1040:
1039:
1019:
1017:
1016:
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1002:
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992:
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951:
950:
938:
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919:
918:
906:
905:
885:
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882:
877:
869:
868:
859:
858:
843:
842:
833:
832:
820:
819:
810:
809:
788:
787:
775:
774:
756:
755:
743:
742:
721:does not occur:
720:
713:
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710:
705:
693:
690:
664:
662:
661:
656:
644:
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615:
613:
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568:
541:
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437:
370:
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362:
354:
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338:
337:
316:
315:
303:
302:
265:
263:
262:
259:{\displaystyle }
257:
252:
251:
239:
238:
172:projective space
138:projective plane
111:
104:
100:
97:
91:
89:
48:
24:
16:
2583:
2582:
2578:
2577:
2576:
2574:
2573:
2572:
2553:
2552:
2551:
2542:
2514:
2505:Delta invariant
2476:
2445:
2409:
2370:AbelâJacobi map
2351:
2325:
2321:Torelli theorem
2291:Dessin d'enfant
2271:Belyi's theorem
2257:
2243:PlĂŒcker formula
2174:
2165:Hurwitz surface
2134:
2113:
2047:
2021:Analytic theory
2013:Elliptic curves
2007:
1988:Projective line
1975:Rational curves
1969:
1963:
1933:
1932:
1925:
1909:
1905:
1897:
1893:
1888:
1850:Algebraic curve
1846:
1793:ramified covers
1708:
1640:
1537:algebraic curve
1533:
1522:
1506:
1498:
1484:
1482:
1464:
1452:
1397:
1394:
1393:
1353:
1308:
1299:
1285:
1277:
1271:
1256:
1250:
1237:
1195:
1179:
1129:
1123:
1118:
1090:
1086:
1077:
1073:
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1054:
1048:
1044:
1035:
1031:
1026:
1023:
1022:
998:
994:
988:
984:
975:
971:
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961:
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942:
933:
929:
914:
910:
901:
897:
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888:
864:
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854:
850:
838:
834:
828:
824:
815:
811:
805:
801:
783:
779:
770:
766:
751:
747:
738:
734:
729:
726:
725:
718:
689:
671:
668:
667:
640:
622:
619:
618:
587:
567:
565:
562:
561:
506:
503:
502:
438:
433:
432:
413:
409:
407:
404:
403:
377:
349:
345:
333:
329:
311:
307:
298:
294:
289:
286:
285:
269:of elements of
247:
243:
234:
230:
225:
222:
221:
197:
124:projective line
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
2581:
2571:
2570:
2565:
2548:
2547:
2544:
2543:
2541:
2540:
2535:
2530:
2524:
2522:
2520:Vector bundles
2516:
2515:
2513:
2512:
2507:
2502:
2497:
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2486:
2484:
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2475:
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2453:
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2402:
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2255:
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2235:
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2215:
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2200:
2195:
2190:
2184:
2182:
2176:
2175:
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2157:
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2146:
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2139:
2136:
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2115:
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2111:
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2101:
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2071:
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2061:
2055:
2053:
2049:
2048:
2046:
2045:
2040:
2035:
2030:
2024:
2022:
2015:
2009:
2008:
2006:
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2000:
1998:Riemann sphere
1995:
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1971:
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1962:
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1923:
1903:
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1882:
1877:
1872:
1867:
1862:
1857:
1852:
1845:
1842:
1832:
1831:
1804:rational curve
1778:crosses itself
1776:where a curve
1638:
1599:function field
1532:
1529:
1520:
1504:
1494:
1478:
1449:
1448:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1351:
1307:
1306:Symmetry group
1304:
1289:
1288:
1281:
1246:
1236:
1233:
1208:Riemann sphere
1194:
1191:
1135:is called the
1125:Main article:
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1114:
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1101:
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1080:
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773:
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746:
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733:
715:
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703:
700:
697:
687:
684:
681:
678:
675:
665:
654:
651:
648:
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635:
632:
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626:
616:
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599:
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591:
585:
582:
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542:
531:
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522:
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516:
513:
510:
484:
483:
472:
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464:
461:
458:
455:
452:
449:
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441:
436:
431:
428:
425:
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419:
416:
412:
383:extended by a
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371:
360:
357:
352:
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336:
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328:
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255:
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229:
196:
193:
114:
113:
28:
26:
19:
9:
6:
4:
3:
2:
2580:
2569:
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2564:
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2560:
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2539:
2536:
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2531:
2529:
2526:
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2521:
2517:
2511:
2508:
2506:
2503:
2501:
2498:
2496:
2493:
2491:
2488:
2487:
2485:
2483:
2482:Singularities
2479:
2473:
2470:
2468:
2465:
2463:
2460:
2458:
2455:
2454:
2452:
2448:
2442:
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2434:
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2427:
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2361:
2358:
2354:
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2343:
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2338:
2335:
2334:
2332:
2330:Constructions
2328:
2322:
2319:
2317:
2314:
2312:
2309:
2307:
2304:
2302:
2301:Klein quartic
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2281:Bolza surface
2279:
2277:
2276:Bring's curve
2274:
2272:
2269:
2268:
2266:
2264:
2260:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
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2234:
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2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2208:Conic section
2206:
2204:
2201:
2199:
2196:
2194:
2191:
2189:
2188:AF+BG theorem
2186:
2185:
2183:
2181:
2177:
2171:
2168:
2166:
2163:
2161:
2158:
2156:
2153:
2151:
2148:
2147:
2145:
2141:
2131:
2128:
2126:
2123:
2122:
2120:
2116:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2056:
2054:
2050:
2044:
2041:
2039:
2036:
2034:
2031:
2029:
2026:
2025:
2023:
2019:
2016:
2014:
2010:
2004:
2003:Twisted cubic
2001:
1999:
1996:
1994:
1991:
1989:
1986:
1984:
1981:
1980:
1978:
1976:
1972:
1968:
1960:
1955:
1953:
1948:
1946:
1941:
1940:
1937:
1926:
1924:9780387977164
1920:
1916:
1915:
1907:
1900:
1895:
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1881:
1878:
1876:
1873:
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1868:
1866:
1863:
1861:
1858:
1856:
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1851:
1848:
1847:
1841:
1839:
1838:
1837:Twisted cubic
1829:
1828:
1827:
1825:
1821:
1817:
1813:
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1800:
1798:
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1790:
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1426:
1423:
1420:
1414:
1408:
1405:
1402:
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1381:
1377:
1373:
1369:
1365:
1361:
1357:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1303:
1297:
1296:
1284:
1280:
1274:
1269:
1268:
1267:
1265:
1259:
1255:elements has
1254:
1249:
1245:
1242:
1232:
1230:
1226:
1222:
1218:
1214:
1210:
1209:
1204:
1200:
1199:complex plane
1190:
1188:
1183:
1178:
1174:
1170:
1167:
1163:
1159:
1154:
1152:
1148:
1147:
1142:
1138:
1134:
1128:
1099:
1091:
1087:
1083:
1078:
1074:
1067:
1062:
1059:
1049:
1045:
1041:
1036:
1032:
1021:
1007:
999:
995:
989:
985:
981:
976:
972:
966:
962:
955:
947:
943:
939:
934:
930:
923:
915:
911:
907:
902:
898:
887:
873:
865:
861:
855:
851:
847:
839:
835:
829:
825:
821:
816:
812:
806:
802:
792:
784:
780:
776:
771:
767:
760:
752:
748:
744:
739:
735:
724:
723:
722:
698:
695:
682:
676:
673:
666:
652:
649:
646:
633:
627:
624:
617:
603:
600:
597:
589:
583:
577:
572:
569:
560:
559:
558:
556:
552:
548:
529:
523:
520:
517:
511:
501:
500:
499:
497:
493:
489:
470:
466:
462:
459:
456:
453:
447:
439:
429:
423:
420:
417:
410:
402:
401:
400:
398:
394:
390:
386:
382:
358:
350:
346:
342:
339:
334:
330:
326:
320:
312:
308:
304:
299:
295:
284:
283:
282:
280:
276:
272:
248:
244:
240:
235:
231:
220:
219:
218:
216:
215:
210:
206:
202:
192:
191:for details.
190:
189:
184:
180:
175:
173:
169:
165:
161:
157:
153:
149:
146:
141:
139:
135:
134:
129:
125:
121:
110:
107:
99:
96:December 2009
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: â
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
2467:Prym variety
2441:Stable curve
2431:Hodge bundle
2421:ELSV formula
2223:Fermat curve
2180:Plane curves
2143:Higher genus
2118:Applications
2043:Modular form
1987:
1913:
1906:
1894:
1880:Wheel theory
1835:
1833:
1823:
1803:
1801:
1786:
1782:ramification
1777:
1773:double point
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1746:). Assuming
1743:
1739:
1735:
1731:
1729:
1710:
1704:
1700:
1696:
1692:
1688:
1684:
1681:rational map
1672:
1668:
1664:
1660:
1653:
1649:
1647:
1642:
1634:
1630:
1626:
1618:
1614:
1606:
1602:
1596:
1591:
1587:
1583:
1579:
1572:
1564:
1556:
1549:non-singular
1544:
1540:
1534:
1524:
1508:
1500:
1495:
1491:
1485:
1479:
1475:
1469:
1458:
1454:
1450:
1383:
1379:
1375:
1371:
1367:
1363:
1360:Transitivity
1359:
1355:
1343:
1339:
1332:group action
1327:
1323:
1319:
1316:coefficients
1312:homographies
1309:
1294:
1290:
1282:
1278:
1272:
1263:
1257:
1252:
1247:
1243:
1241:finite field
1238:
1213:Gauss sphere
1212:
1206:
1196:
1185:Compare the
1184:
1173:group theory
1157:
1155:
1150:
1144:
1140:
1136:
1133:real numbers
1130:
716:
554:
550:
546:
544:
495:
491:
487:
485:
396:
392:
388:
380:
378:
278:
270:
268:
212:
204:
200:
198:
186:
176:
168:vector space
163:
155:
151:
147:
142:
131:
123:
117:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
2342:Polar curve
1855:Cross-ratio
1517:cross-ratio
1295:at infinity
1166:identifying
1162:unit circle
399:) given by
120:mathematics
2557:Categories
2337:Dual curve
1965:Topics in
1886:References
1338:, so that
1336:transitive
275:equivalent
66:newspapers
2450:Morphisms
2198:Bitangent
1551:curve of
1488:= 1, 2, 3
1430:λ
1421:λ
1415:∼
1270:for each
1164:and then
1160:onto the
1060:−
924:⋅
702:∞
686:∞
680:∞
637:∞
631:∞
628:⋅
593:∞
581:∞
509:∞
460:∈
454:∣
430:∈
343:λ
327:λ
321:∼
160:subspaces
1844:See also
1818:is 0. A
1656:) of an
1567:, up to
1472:distinct
1384:artifact
1330:). This
1177:subgroup
1116:Examples
699:≠
650:≠
183:manifold
2510:Tacnode
2495:Crunode
1814:); its
1633:) over
1547:) is a
1474:points
1346:) is a
1180:{1, â1}
80:scholar
2490:Acnode
2414:Moduli
1921:
1621:. The
1555:0. If
1370:. The
1293:point
1203:sphere
185:; see
82:
75:
68:
61:
53:
1816:genus
1683:from
1663:over
1613:over
1609:) of
1577:conic
1575:to a
1553:genus
1314:with
181:is a
179:reals
145:field
87:JSTOR
73:books
2500:Cusp
1919:ISBN
1834:See
1709:dim
1597:The
1483:for
1223:and
128:line
122:, a
59:news
1762:to
1730:If
1719:of
1713:â 1
1687:to
1625:of
1559:is
1374:on
1334:is
1318:in
1276:in
1260:+ 1
1251:of
549:to
211:of
118:In
42:by
2559::
1802:A
1784:.
1457:,
1302:.
1231:.
1219:,
1182:.
691:if
642:if
498::
281::
174:.
1958:e
1951:t
1944:v
1928:.
1830:.
1824:P
1768:K
1766:(
1764:P
1760:C
1756:C
1754:(
1752:K
1748:C
1744:K
1742:(
1740:P
1736:C
1732:V
1711:V
1705:P
1701:K
1699:(
1697:P
1693:K
1691:(
1689:P
1685:V
1673:T
1671:(
1669:K
1665:K
1661:V
1654:V
1652:(
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1627:K
1619:T
1615:K
1607:T
1605:(
1603:K
1592:P
1588:K
1584:C
1580:C
1573:K
1565:K
1557:K
1545:K
1543:(
1541:P
1525:K
1523:(
1521:2
1509:K
1507:(
1505:2
1496:i
1492:R
1486:i
1480:i
1476:Q
1461:)
1459:Y
1455:X
1453:(
1436:]
1433:Y
1427::
1424:X
1418:[
1412:]
1409:Y
1406::
1403:X
1400:[
1380:K
1378:(
1376:P
1368:R
1364:Q
1356:K
1354:(
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1342:(
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1320:K
1287:,
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1273:a
1264:q
1258:q
1253:q
1248:q
1244:F
1158:R
1151:K
1141:K
1100:.
1097:]
1092:1
1088:x
1084::
1079:2
1075:x
1071:[
1068:=
1063:1
1056:]
1050:2
1046:x
1042::
1037:1
1033:x
1029:[
1008:,
1005:]
1000:2
996:y
990:2
986:x
982::
977:1
973:y
967:1
963:x
959:[
956:=
953:]
948:2
944:y
940::
935:1
931:y
927:[
921:]
916:2
912:x
908::
903:1
899:x
895:[
874:,
871:]
866:2
862:y
856:2
852:x
848::
845:)
840:2
836:x
830:1
826:y
822:+
817:2
813:y
807:1
803:x
799:(
796:[
793:=
790:]
785:2
781:y
777::
772:1
768:y
764:[
761:+
758:]
753:2
749:x
745::
740:1
736:x
732:[
696:x
683:=
677:+
674:x
653:0
647:x
634:=
625:x
604:,
601:0
598:=
590:1
584:,
578:=
573:0
570:1
555:K
553:(
551:P
547:K
530:.
527:]
524:0
521::
518:1
515:[
512:=
492:K
490:(
488:P
471:.
467:}
463:K
457:x
451:)
448:K
445:(
440:1
435:P
427:]
424:1
421::
418:x
415:[
411:{
397:K
395:(
393:P
389:K
381:K
359:.
356:]
351:2
347:x
340::
335:1
331:x
324:[
318:]
313:2
309:x
305::
300:1
296:x
292:[
279:λ
271:K
254:]
249:2
245:x
241::
236:1
232:x
228:[
205:K
203:(
201:P
166:-
164:K
156:K
154:(
152:P
148:K
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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