THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES PREDEGREE POLYNOMIALS OF PLANE CONFIGURATIONS IN PROJECTIVE SPACE DIMITRE G.

Transcription

1 THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES PREDEGREE POLYNOMIALS OF PLANE CONFIGURATIONS IN PROJECTIVE SPACE By DIMITRE G. TZIGANTCHEV A Dssertaton submtted to the Department of Mathematcs n partal fulfllment of the requrements for the degree of Doctor of Phlosophy Degree Awarded: Fall Semester, 006

2 The members of the Commttee approve the Dssertaton of Dmtre G. Tzgantchev defended on October 30, 006. Paolo Aluff Professor Drectng Dssertaton Laura Rena Outsde Commttee Member Ettore Aldrovand Commttee Member Erc Klassen Commttee Member Mka Seppälä Commttee Member The Offce of Graduate Studes has verfed and approved the above named commttee members.

3 To Mlena

4 ACKNOWLEDGEMENTS The author s grateful to all members of hs ATE commttee, namely Dr. Aluff, Dr. Aldrovand, Dr. Klassen, Dr. Hronaka and Dr. Seppälä, for ther comments and suggestons on ths dssertaton. Many thanks to our departmental char Dr. Bowers and the outsde commttee member Dr. Rena as well. Most specal and sncere thanks go to Dr. Aluff for hs proposal of the problem of ths Ph.D. thess and hs contnuous and vtal support n the process of preparng t. Hs help n provng Theorem.6.4, for example, s greatly apprecated. The computatonal ngredent of ths dssertaton was substantal. The (arthmetc) calculatons alone have taken many months. The author thanks hs wfe Mlena and hs daughter Adrana for ther patence, encouragement and belef n hm as most of the above mentoned work was done at home. Last, but certanly not least, Dmtre thanks hs mother and parents-n-law for ther crossng fngers overseas. Regardng hs teachng career and overall experence at FSU, the author also acknowledges Dr. Stles and Susan Mnnerly for ther professonalsm. v

5 TABLE OF CONTENTS Abstract v 1. INTRODUCTION THE PROBLEM. GEOMETRY Orbt Geometrc Interpretaton Lnear Systems. Pont Condtons Base Locus Blow-Ups Proof of Geometrc Interpretaton Predegrees INTERSECTION THEORY Contrbutons Frst Contrbuton Second Contrbuton Thrd Contrbuton PREDEGREE POLYNOMIAL Up to Degree Eght Degree Nne Degree Thrteen TRANSVERSAL CONFIGURATIONS Transton from C to S Transton from K to Q SPECIAL CONFIGURATIONS. CONCLUSIONS Specal Confguratons Future Work APPENDICES A. THIRD CONTRIBUTIONS v

6 B. PRODUCT OF PLANE POLYNOMIALS C. POWERS OF d D. C-, K- FUNCTIONS E. MAPLE CODES REFERENCES BIOGRAPHICAL SKETCH v

7 ABSTRACT We work over an algebracally closed ground feld of characterstc zero. The group of P GL(4) acts naturally on the projectve space P N parameterzng surfaces of a gven degree d n P 3. The orbt of a surface under ths acton s the mage of a ratonal map P GL(4) P 15 P N. The closure of the orbt s a natural and nterestng object to study. Its predegree s defned as the degree of the orbt closure multpled by the degree of the above map restrcted to a general P j, j beng the dmenson of the orbt. We fnd the predegrees and other nvarants for all surfaces supported on unons of planes. The nformaton s encoded n the so-called adjusted predegree polynomals, whch possess nce multplcatve propertes allowng us to easly compute the predegree (polynomals) of varous specal plane confguratons. The predegree has both a combnatoral and geometrc sgnfcance. The results obtaned n ths thess would be a necessary step n the soluton of the problem of computng predegrees for all surfaces. v

8 CHAPTER 1 INTRODUCTION The group P GL(3) of projectve lnear transformatons of P acts naturally on the space P d(d+3)/ parameterzng plane curves of degree d. Aluff and Faber ([1]) have computed the degree of the closure n ths space of the orbt of an arbtrary plane curve. The orbt closure of a curve s a natural object of study and ts degree has a smple enumeratve meanng: for a reduced curve wth fnte stablzer, t counts the number of translates of the curve that contan eght gven general ponts. For a nonsngular curve ths s, roughly speakng, one of the Gromov-Wtten nvarants of P. The case of curves supported on unons of lnes s of specal separate nterest ([], [3]). The constructons n ths case were essental for the computatons n the general case. We have a smlar stuaton for surfaces n P 3. We use P N = P d(d +6d+11)/6 to parameterze degree d surfaces n P 3. The group P GL(4) acts naturally on P N and we can ask the same queston of computng the degree of the closure n ths space of the orbt of a surface. It stll has a smlar geometrc meanng: for a reduced surface wth fnte stablzer, t counts the number of translates of the surface that contan ffteen gven general ponts. More generally, f j s the dmenson of the closure of the orbt of a reduced surface, then the degree of the orbt s the number of translates contanng j ponts n general poston. In very specal cases (for example when the surface conssts of 3 planes n general poston) ths number can be computed by nave combnatoral consderatons. In general ths s not possble. We consder the case of surfaces supported on unons of planes,.e. plane confguratons. Most of the constructons n [3] are easly carred out n ths case. The acton of P GL(4) defnes a regular map s : P GL(4) P N for a fxed plane confguraton S n P N. In terms of ths map then the orbt of S s Im(s), denoted also O S. The map s extends to a ratonal map from the compactfcaton P 15 of P GL(4) to P N whch s also denoted by s. 1

9 We defne the (adjusted) predegree polynomal of S to be 0 (f d )t /!, where f = degs P = the number of ponts n a general fber of s P and d s the ntersecton number of s(p ) and codmenson lnear subspace of P N for a general P n P 15. The denomnators are ntroduced n the defnton to gve the polynomal a certan structure whch would be otherwse lost. Here are a few more useful observatons regardng ths defnton. If j =dmo S, then d k = 0 for k > j and the degree of the polynomal s j 15, d k =degs(p k ) for k < j, d j s the degree of the orbt closure, f j s the degree of the stablzer of S and f k = 1 for k < j, f j s the number of rreducble components of the closure of the stablzer of S. The leadng coeffcent f j d j s called the predegree of (the orbt of) S. Ths nformaton records the degree of the orbt closure of S, corrected by the sze of the stablzer of the confguraton. The predegree retans the enumeratve nterpretaton mentoned above. For example, f j =dmo S and S s reduced, then the predegree for S s the number of ordered plane confguratons n O S whch go through j general ponts. Geometrc sense can be made also n the case of nonreduced confguratons. Arrangements of planes are nterestng from varous vewponts (e.g. secton 6.). From the pont of vew of ntersecton theory, the predegree s a more natural (and more straght forwardly computable) nvarant than the degree of the orbt closure. Thus as n the case of lne confguratons n the plane, the basc queston of fndng the degree s reduced to the queston of fndng the predegree for a gven plane confguraton n P 3. Also, n vew of the fnal expresson we fnd for the predegree polynomals, we are nterested n the whole polynomal rather than the predegree of the plane confguraton alone. It allows us to deal at once wth all confguratons of planes, regardless of the dmenson of ther stablzer (whch s 15 mnus the degree of the polynomal). It also carres more nvarants of the orbt closure such as ts dmenson and the degrees of subsets of the orbt closure determned by mposng lnear condtons on the transformaton appled to the gven surface (degs(p ) for a general projectve lnear subspaces P of P 15 ). The man result of ths dssertaton s the computaton of the predegree polynomal of an (arbtrary) plane confguraton S (wth possbly multple planes) by the means of a sequence of blow-ups over P 15. We construct a regular map s : Ṽ PN whch lfts the ratonal map s

10 and resolves ts ndetermnaces (see the dagram below). An ntersecton-theoretc analyss of ths map allows us to compute the coeffcents of the predegree polynomal for S. Ṽ s π P 15 s P N (π s the composton of the blow-ups) Agan, n an ntersecton-theoretc framework, the predegree polynomal of S may be shown to equal H 15 W 0 t /!, where H s the pull-back to Ṽ of the hyperplane class Ṽ H n P 15 and W s the pull-back to Ṽ of the class W of specal hyperplanes n PN called pont condtons (correspondng to surfaces n P N gong through a fxed pont). The man tool n the computaton of the above ntersecton numbers, appearng as coeffcents n the predegree polynomal, s Theorem II from [4]. It allows us to control the behavor of ntersecton numbers through blow-ups. Agan, as n the case of lnes n the plane, the computatons fnally amount to fndng ntersecton products of push-forwards of ntersecton classes whch we obtan by usng the b-ratonal nvarance of Segre classes of normal cones and other tools from Fulton-MacPherson s ntersecton theory ([5]). The coeffcents turn out to be polynomal functons dependng on the multplctes of the planes, lnes and ponts of ntersecton (.e., the multplctes of S along ts plane, lne and pont components). Because of the large expressons we have to deal wth, when computng the hgher degree coeffcents, the use of a symbolc computaton package such as Maple s nevtable. The bgger problem however s makng a geometrc sense out of the raw coeffcents (computed by Maple). We manage to put them n a more useful form. It transparently accounts for the combnatorcs of the arrangement of planes. The result s that the predegree polynomal can be wrtten as the truncaton to t 15 of the sum of three contrbutons where the frst one uses the data of the number of planes and ther multplctes; the second one n addton to ths needs nformaton about the multplctes of the lnes of ntersecton of the planes and the last one also depends on the multplctes of all of these - planes, lnes and ponts of ntersecton,.e. the last two terms use ncdence data unlke the frst one. 3

11 A crucal observaton then s that the second and the thrd contrbutons vansh for specfc confguratons. Namely, the second term s zero f no plane contans a lne of ntersecton of the rest of the confguraton and the last one s zero f the plane confguraton s transversal. In partcular, the predegree polynomal of n general planes s gven by the frst contrbuton only and t s the truncaton of the product n =0 (1 + r t + r t / + r 3 t 3 /3!) of the predegree polynomals of the planes. Another extremely useful characterstc of the predegree polynomal s the structure of the contrbutons themselves. Beng combnatons of truncatons of Taylor expansons of exponental functons, they enjoy good multplcatvty propertes. Ths fact provdes for a nce multplcaton rule of whole polynomals for certan confguratons. In these cases the predegree polynomal of the unon of two such confguratons s smply (the truncaton of) the product of the polynomals of the confguratons f the latter meet transversally. Ths fact generalzes to plane arrangements a phenomenon whch was dscovered n the case of lne confguratons. In the latter case, however, the transversalty of two confguratons s the only requrement for the product of the two polynomals to be the predegree polynomal of the unon of the confguratons. The case of plane arrangements s a bt subtler and the addtonal condton we need for planes s the vanshng of the mddle contrbuton term whch we saw has to do wth the lnes, the only proper nonzero dmensonal components of ntersecton. These non-obvous propertes of predegree polynomals together wth other observatons (for example about the stablzer of some confguratons) endow the theory wth a non-trval structure, whch would be nterestng to study further. As an applcaton, smple formulae may be found for specfc arrangements of planes whch we call books, stars, fans, etc. n agreement wth the correspondng termnology for lnes n the plane. References to standard texts wll be gven for the facts used n ths thess. For example, the reader may fnd most of the prelmnares needed n the next two chapters of the thess n chapters 1-4 of [5] (n case a specfc reference s not present). 4

12 CHAPTER THE PROBLEM. GEOMETRY. In ths chapter we show how and why the man problem of fndng the predegree polynomal of a plane confguraton n P 3 evolves from the orgnal one of fndng the degree of the closure of the lnear orbt of the confguraton n the projectve space parameterzng all surfaces and n partcular all plane confguratons of a gven degree. Alongsde we present all prelmnares n the order they are needed for our consderatons..1 Orbt Let P 3 be the three dmensonal projectve space over a fxed algebracally closed feld K of characterstc zero. Let us choose projectve coordnates (x : y : z : w) n P 3 and consder algebrac surfaces n P 3 of fxed degree d. Each such surface S s defned as the zero set of a degree d homogeneous polynomal F (x : y : z : w) = +j+k+l=d a jklx y j z k w l. The number of all words (monomals) x y j z k w l of total degree d n x, y, z, w can be computed usng standard combnatoral arguments. It s d(d + 6d + 11)/6 + 1, so we can use P N = P d(d +6d+11)/6 (wth coordnates a jkl ) to parameterze degree d surfaces n P 3. Examples.1.1 () When d =, P N = P 9 parameterzes all quadrcs, because there are ten words n x, y, z, w of second degree: x, y, z, w, xy, yz, zw, xw, yw, xz and each pont (a 0 : a 1 : : a 9 ) P 9 can be dentfed wth the polynomal F = a 0 x +a 1 y +a z +a 3 w +a 4 xy +a 5 yz + a 6 zw + a 7 xw + a 8 yw + a 9 xz whch n turn determnes a unque quadrc surface S n P 3. () If d = 1, then P N = ˇP 3, the dual space of P 3, consstng of all planes n P 3. The projectve general lnear group P GL(4) of 4 4 matrces over K acts on P N n a 5

13 natural way. Letφ = φ 0 φ 1 φ φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 10 φ 11 φ 1 φ 13 φ 14 φ 15 P GL(4) and a surface S (as an element of P N ) be defned by F (x : y : z : w) = 0. Then we defne the acton of φ on S, denoted S φ, to be the surface wth equaton F (φ(x : y : z : w)) = 0,.e., F (φ 0 x + φ 1 y + φ z + φ 3 w : φ 4 x + φ 5 y + φ 6 z + φ 7 w : φ 8 x + φ 9 y + φ 10 z + φ 11 w : φ 1 x + φ 13 y + φ 14 z + φ 15 w) = 0. Let us now fx a surface S and defne a regular map s from P GL(4) to P N nduced by the P GL(4) acton on S. More precsely, whch s s : P GL(4) P N, φ S φ The mage Im(s) of the map s s called the lnear orbt of the surface S because t s the orbt of S under the P GL(4) acton. It s denoted by O S and conssts of all translates of S. Next, we thnk of the ponts of a P 15 as of 4 4 matrces modulo scalar multplcaton, so we can embed P GL(4) n P 15 wth coordnates (φ 0 :... : φ 15 ). P GL(4) s an open set of P 15 because t s the complement of the closed set {φ P 15 / det(φ) = 0}. Ths embeddng P GL(4) P 15 extends the regular map s to a ratonal map s : P 15 P N whch we stll denote by s because t s defned the same way nsde and wherever possble outsde P GL(4). Then the orbt closure O S mage of ths ratonal map s. of S s the closure of the Denote, as usual, the stablzer of S under the P GL(4) acton by Stab S. In other words, ths s the subgroup of P GL(4) whch s the fber s 1 (S) over S. The dmenson of a general fber of s equals dm Stab S. Snce dm P GL(4) = dm P 15 = 15, then dm O S + dm Stab S = 15. In partcular, dm O S 15. We say that the orbt O S s small when ts dmenson s less than the maxmal possble 15,.e., when Stab S has postve dmenson. From now on the surface S wll be supported on unons of planes,.e., we wll consder plane confguratons (or arrangements) only, although some of the results (e.g. about pont condtons n P 15 ) wll hold for surfaces n general. In ths case the defnng polynomal F of S wll be a product of lnear factors correspondng to the planes,.e., F (x : y : z : w) = n =1 Lr (x : y : z : w) where r s the multplcty of the -th plane, L = 0 s the equaton 6

14 of the -th plane and n s the number of all planes (counted wth multplcty). We can also wrte that S = n =1 P where P s the plane determned by L. The closure of the lnear orbt of S s a natural and nterestng object of study and one of ts most mportant characterstcs s ts degree deg O S. It possesses a nce enumeratve geometrc nterpretaton llustrated n the next secton.. Geometrc Interpretaton In ths secton we present our man motvaton for studyng the orbt closure of a plane confguratons. The proof of the theorem below s gven n secton.6. and t uses materal from sectons before t. Theorem..1 If the dmenson of the orbt of a reduced plane confguraton s j, then the degree of the orbt closure equals the number of translates of the plane confguraton whch pass through j general ponts. As an applcaton of ths theorem, we see that n very specal cases ths number can be computed by nave combnatoral consderatons. These are the cases n whch the orbt conssts of all confguratons wth the same ncdence data as the orgnal confguraton. In general ths s not possble. In the remander of ths secton we llustrate the combnatoral case wth the followng examples. Example.. If a plane confguraton S s a sngle reduced plane, then clearly ts orbt conssts of all reduced planes n P 3. So ts dmenson should be 3 and there s only one plane contanng 3 general ponts,.e., deg O S = 1. Example..3 Let now S be a confguraton consstng of two dstnct reduced planes. The orbt conssts of all pars of planes and hence ts dmenson s 6 (3 degrees of freedom for each plane). Thus we need to count how many pars of planes contan 6 ponts n general poston n P 3. If 3 of the ponts determne a plane, then the other 3 ponts automatcally determne the other plane. 7

15 Hence the number of pars of planes gong through 6 general ponts s ( 6 3) / = 10, because ) counts each par twce. Therefore, the degree of the orbt closure deg OS s 10. ( 6 3 Example..4 Naturally, our next example wll be of a confguraton S of three general (not gong through a lne) reduced planes. Agan, the orbt conssts of all confguratons lke the orgnal one,.e., of confguratons of 3 general reduced planes. Then the dmenson of the orbt s 9 and lke n the prevous example we count the number of all such confguratons through 9 general ponts to be ( 9 3)( 6 3) /6 = deg OS = 80. Example..5 Ths s the frst example n whch the plane confguraton S does not consst of general planes. Consder three reduced planes wth a common lne. We call ths confguraton a 3-book. Its orbt conssts of all such 3-books n P 3 and has dmenson 7 (3 degrees of freedom for each of planes, 1 degree of freedom for the plane that completes the 3-book and contans the ntersecton of the frst two planes). There are two ways n whch such a confguraton can contan 7 general ponts. Frst, f each of two of the planes contans 3 ponts, then the thrd plane (hence the whole confguraton) s determned by the remanng pont. The number of such confguratons s ( 7 3)( 4 3) / = 70. We dvde by because otherwse the par of the frst two planes would be counted twce. The other possblty occurs when one plane s determned by 3 of the ponts and of the remanng 4 ponts we choose pars of ponts. Each par determnes a lne ntersectng the plane n a pont. The lne n the plane whch connects these two ponts s the lne (spne) of the 3-book,.e., the other two planes are unquely determned f we requre that each of them contan a par of ponts and that all three planes form a 3-book. 8

16 Clearly the computaton n ths case s ( 7 )( 5 ) / = 105 (the dvson by s because of the same reason as above). So the total number of confguratons s = 175 = deg O S. Example..6 Buldng up on the prevous example, the next confguraton wll be the unon S of 3 reduced planes wth a common lne and one more reduced general plane (whch does not contan the lne),.e., a 3-book plus a general plane. Its orbt conssts of all such confguratons n P 3 and has dmenson 10 (3 degrees of freedom for each of planes, 1 degree of freedom for the plane that completes the 3-book and contans the ntersecton of the frst two planes, and 3 degrees of freedom for the last general plane). We can now use Example..5 to compute the number of confguratons that contan 10 general ponts. There are ( 10 3 ) = 10 ways to choose the general plane and 175 possbltes for the 3-book accordng to the prevous example. Thus the total number of confguratons s = 1000 = deg O S. Remark..7 Besdes the confguraton of four general planes, whch can be done n a manner smlar to Examples..3 and..4, there s one more confguraton whch depletes all possbltes of plane confguratons of up to four planes, namely the 4-book. Ths however s our frst non-combnatoral example because the dmenson of the set of all 4-books s clearly 8, whereas, as we wll see n Theorem 6.1.5, the dmenson of the orbt of an n-book s 7. In other words, the orbt of a 4-book does not consst of all 4-books n P 3. In secton.6. we wll demonstrate how we can make use of Theorem..1. even for nonreduced plane confguratons. 9

17 .3 Lnear Systems. Pont Condtons Let f : P m P n be a ratonal map between two projectve spaces,.e., a map defned by f(x 0 : x 1 : : x m ) = (f 0 (x 0 : x 1 : : x m ) : f 1 (x 0 : x 1 : : x m ) : : f n (x 0 : x 1 : : x m )), where (x 0 : x 1 : : x m ) s a pont of P m and f 0, f 1,..., f n are homogeneous polynomals of the same degree. By defnton, the lnear system defned by f s span{f 0, f 1,..., f n }. In fact, not only s the lnear system determned by f, but n a sense f s determned by the lnear system,.e., there s a one-to-one correspondence between ratonal maps and lnear systems (pg.156 [6]). By ths we mean that f the same lnear system s spanned by the same mnmal number of n other generators, then clearly they determne the same map f up to a change of coordnates n P n. Also, a lnear system, obtaned from another lnear system after removng any possble fxed common components of the elements of ths system, defnes the same ratonal map. The two lnear systems smply defne two representatves of the equvalence class called ratonal map. The base locus of f s n =0 dv(f ),.e., f s a regular map f and only f ts base locus s empty. In vew of ths and the above observatons, the base locus of f can be thought as of the ntersecton of (the dvsors of the elements of) any (not necessarly mnmal) generatng set of the lnear system or just as the ntersecton of the whole lnear system of f. Let P n have coordnates (y 0 : y 1 : : y n ). Then the hyperplane y = 0 n P n pulls back to the hypersurface f (x 0 : x 1 : : x m ) = 0 n P m, 0 n. So the lnear system of f conssts of the pull-backs of the hyperplanes n P n. Also, the noton of a lnear system extends to any (ratonal) map f : X P n to a projectve space P n (from any source X) and t s stll true that, up to change of coordnates n P n, the set of all such maps f s n one-to-one correspondence wth the set of the correspondng lnear systems (whose elements have no fxed common components). lnear systems, n turn, consst of the pull-backs of the hyperplanes n P n. The base locus s defned the same way as the ntersecton of these pull-backs and the map f s regular f and only f the base locus s empty. Let us now concentrate on the case n whch the target P n of the map f s P N, the space parameterzng surfaces of fxed degree n P 3 (we wll specfy the source later). The coordnates n P N wll be (y 0 : y 1 : : y N ). The 10

18 All surfaces n P 3 gong through a fxed pont clearly form a hyperplane n P N. Such hyperplanes are called pont condtons. Each pont condton corresponds to a pont n ˇP N, the dual space of P N. Next, we realze the set of all pont condtons n ˇP N as the mage of the Veronese map v : P 3 ˇ P N gven by v(x : y : z : w) = (v 0 : v 1 : : v N ) where v 0, v 1,..., v N are all monomals x y j z k w l of degree d n x, y, z, w (n some prescrbed order, for example the lexcographcal one). Indeed, to see that (v 0 : v 1 : : v N ) corresponds to all degree d surfaces whch go through the pont (x : y : z : w) we smply follow defntons. The pont (v 0 : v 1 : : v N ) n ˇP N corresponds to the hyperplane n P N wth coeffcents (v 0 : v 1 : : v N ),.e., the hyperplane consstng of all ponts (y 0 : y 1 : : y N ) P N for whch v 0 y 0 +v 1 y 1 + +v N y N = 0. Fnally, accordng to the constructon of P N, ths amounts to havng all surfaces wth equatons N =0 y v = 0,.e., all degree d surfaces passng through (x : y : z : w). The crucal observaton about ths mage s that t s non-degenerate,.e., not contaned n any hyperplane of ˇP N. Ths s so because no nontrval lnear combnaton of v 0, v 1,..., v N s dentcally zero. The geometrc pont of vew s also clear: f the set of pont condtons s degenerate n ˇP N, then the pont condtons n P N, whch are hyperplanes, should all go through one pont, whch s not true, as ths pont would correspond to a surface contanng all ponts of P 3. Snce the span of a non-degenerate set s a non-degenerate lnear subspace of ˇP N, then t must be the whole space and we have just proved Proposton.3.1 The pont condtons span ˇP N. In other words, every hyperplane n P N s a lnear combnaton of pont condtons. Let us now return to our map s : P 15 P N. We defne the pont condtons n P 15 to be the pull-backs of the pont condtons from P N. In summary, the lnear system of s s spanned by the pont condtons n P 15,.e., snce the pont condtons n Im(s) P N consst of translates S φ of S whch go through a fxed pont p, the lnear system of s s span{φ P 15 /S φ p or S φ s undefned}. Also, a pont condton correspondng to a pont p has an equaton F (φ(p)) = 0 n P 15 and the base locus of s s the ntersecton over all ponts n P 3 of the pont condtons n P 15. Thus the base locus s {φ P 15 /F (φ(x : y : z : w) 0}, whch s also clear a pror n vew of the defnton of the ratonal map s extendng the regular map s. We wll use ths descrpton n the next secton to fnd the base locus of s. 11

19 .4 Base Locus Let us recall that we consder the case n whch the support of a surface S s the unon n =1 P of planes P and the defnng equaton of S s F (x : y : z : w) = n (x : y : z : w) where =1 Lr r s the multplcty of the -th plane, L = 0 s the equaton of the -th plane P and n s the number of all (non-reduced) planes. We also remember that the base locus of s s {φ P 15 /F (φ(x : y : z : w) 0}, so t s n =1 {φ P15 /L (φ(x : y : z : w) 0}. Consder any component of ths unon. It conssts of φ 0 φ 1 φ φ 3 all φ = (φ 0 : φ 1 : : φ 15 ) = φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 10 φ 11 for whch φ(p3 ) s n P. In partcular, φ 1 φ 13 φ 14 φ 15 t depends on the plane P only (and not the rest of S). Wthout loss of generalty we may assume that the equaton L = 0 of P s w = 0. Then the correspondng component of the base locus conssts of all φ s for whch φ 1 x + φ 13 y + φ 14 z + φ 15 w s zero for all ponts (x : y : z : w) P 3,.e., t s {φ P 15 /φ 1 = φ 13 = φ 14 = φ 15 = 0} whch, as an ntersecton of four dstnct hyperplanes, s a P 11 n P 15. Thus the base locus of s s a unon of P 11 s whch are n a one-to-one correspondence wth the planes P n the plane confguraton and each such P 11 s the set of all elements of P 15 whch map P 3 nto a lnear subspace of the component P. Next we want to descrbe the ntersectons of these components n a smlar fashon. Clearly, the ntersecton of any two P 11 s conssts of matrces sendng P 3 nto a lnear subspace of the lne whch s the ntersecton of the two planes correspondng to the two P 11 s. If we, agan wthout loss of generalty, assume that these are the planes z = 0 and w = 0, we can show that any two P 11 s ntersect n a P 7. Indeed, the ntersecton set s characterzed by φ 8 x + φ 9 y + φ 10 z + φ 11 w φ 1 x + φ 13 y + φ 14 z + φ 15 w 0, hence t s {φ P 15 /φ 8 = φ 9 = φ 10 = φ 11 = φ 1 = φ 13 = φ 14 = φ 15 = 0} P 7. Smlarly, the ntersecton of any three such dstngushed P 11 s (.e. the ntersecton of any dstngushed P 11 and any dstngushed P 7 ) s a P 3 whch corresponds to the pont of ntersecton of the three planes assocated wth the three P 11 s. To check ths we can safely assume agan that the these three planes are the planes y = 0, z = 0, w = 0 and get that the P 3 s the ntersecton of the hyperplanes φ 4 = φ 5 = φ 6 = φ 7 = φ 8 = φ 9 = φ 10 = φ 11 = φ 1 = φ 13 = φ 14 = φ 15 = 0. Also, any such dstngushed P 3 s the set of matrces n P 15 whch map P 3 to the pont correspondng to the P 3. 1

20 The geometrc nterpretaton of the dstngushed P 11 s, P 7 s and P 3 s mples that no other ntersectons occur n the base locus. Any two P 7 s ether ntersect along a P 3 or are dsjont; a P 3 s ether completely nsde or outsde a P 11 or a P 7 and all P 3 s are dsjont. Fnally, note that all components of the base locus consst of sngular matrces, hence they le n the complement of P GL(4) n P Blow-Ups Ths s the secton n whch we resolve the ndetermnaces of the ratonal map s, whch wll be essental n the next sectons. We do ths by the means of a sequence of blow-ups of P 15 along the dstngushed sets n the base locus of s. More precsely, there are three stages (sequences) of blow-ups and one theorem s devoted to each of them. After a coordnate change n P 3 the pont (1 : 0 : 0 : 0) wll be a pont of ntersecton of some of the planes n the confguraton S,.e., we can assume that the equaton of S s F (x : y : z : w) = (β y + γ z + δ w) r j (α jx + β j y + γ j z + δ j w) r j where α j 0. Accordng to secton.3, the pont condton n P 15 (φ 0 : φ 1 : : φ 15 ) correspondng to a pont (x 0 : y 0 : z 0 : w 0 ) s (β (φ 4 x 0 + φ 5 y 0 + φ 6 z 0 + φ 7 w 0 ) + γ (φ 8 x 0 + φ 9 y 0 + φ 10 z 0 + φ 11 w 0 ) + δ (φ 1 x 0 + φ 13 y 0 + φ 14 z 0 +φ 15 w 0 )) r j (α j(φ 0 x 0 +φ 1 y 0 +φ z 0 +φ 3 w 0 )+β j (φ 4 x 0 +φ 5 y 0 +φ 6 z 0 +φ 7 w 0 )+γ j (φ 8 x 0 + φ 9 y 0 + φ 10 z 0 + φ 11 w 0 ) + δ j (φ 1 x 0 + φ 13 y 0 + φ 14 z 0 + φ 15 w 0 )) r j = 0 where α j 0. P 15. We consder all blow-ups over the representatve affne chart A 15 (1 : p 1 : p : : p 15 ) n Theorem.5.1 Let V be the varety obtaned from V = P 15 after all blow-ups centered at the dstngushed P 3 s (f any) n the base locus of s. (frst blow P 15 up along a P 3, then blow up the blow-up along a proper transform of another P 3 and contnue tll all P 3 s have been used.) Then () The proper transforms of the dstngushed P 7 s n V are dsjont. () The multplcty of a pont condton along each P 3 correspondng to the ntersecton of planes of the confguraton S equals the sum of the multplctes of those planes. () The ntersecton of the proper transforms of the pont condtons n V conssts of the proper transforms of the dstngushed P 11 s. Proof () s clear from the structure of the base locus dscussed n the prevous secton and the fact that transversal ntersectons are separated by blow-ups. 13

21 In A 15 (1 : p 1 : p : : p 15 ) P 15 the pont condton correspondng to (x 0 : y 0 : z 0 : w 0 ) has the equaton (β (p 4 x 0 + p 5 y 0 + p 6 z 0 + p 7 w 0 ) + γ (p 8 x 0 + p 9 y 0 + p 10 z 0 + p 11 w 0 ) + δ (p 1 x 0 + p 13 y 0 + p 14 z 0 + p 15 w 0 )) r j (α j(x 0 + p 1 y 0 + p z 0 + p 3 w 0 ) + β j (p 4 x 0 + p 5 y 0 + p 6 z 0 + p 7 w 0 ) + γ j (p 8 x 0 + p 9 y 0 + p 10 z 0 + p 11 w 0 ) + δ j (p 1 x 0 + p 13 y 0 + p 14 z 0 + p 15 w 0 )) r j = 0 where α j 0. The frst tme we blow-up V along a P 3 we can assume that ths s the P 3 whch corresponds to the pont (1 : 0 : 0 : 0),.e., the P 3 wth equatons p 4 = p 5 = = p 15 = 0 (as found n the prevous secton). The map from a representatve chart A 15 (q 1, q,..., q 15 ) of the blow-up to A 15 (p 1, p,..., p 15 ) n V s gven by the followng equatons: p 1 = q 1, p = q,..., p 4 = q 4, p 5 = q 4 q 5, p 6 = q 4 q 6,..., p 15 = q 4 q 15. In ths A 15 (q 1, q,..., q 15 ) the exceptonal dvsor s q 4 = 0. The (full) transform of the above pont condton then s qr 4 (β (x 0 + q 5 y 0 + q 6 z 0 + q 7 w 0 ) + γ (q 8 x 0 + q 9 y 0 + q 10 z 0 + q 11 w 0 ) + δ (q 1 x 0 + q 13 y 0 + q 14 z 0 + q 15 w 0 )) r j (α j(x 0 + q 1 y 0 + q z 0 + q 3 w 0 ) + β j q 4 (x 0 + q 5 y 0 + q 6 z 0 + q 7 w 0 ) + γ j q 4 (q 8 x 0 + q 9 y 0 + q 10 z 0 + q 11 w 0 ) + δ j q 4 (q 1 x 0 + q 13 y 0 + q 14 z 0 + q 15 w 0 )) r j = 0. Snce α j 0, ths shows that the multplcty of the exceptonal dvsor n ths transform s r, whch proves () for ths partcular P 3 whence for all other dsjont P 3 s. () s true on the complement of the exceptonal dvsors, so we only need to verfy t along each exceptonal dvsor. We fnd the ntersecton of the proper transform of the pont condton wth the exceptonal dvsor: (β (x 0 + q 5 y 0 + q 6 z 0 + q 7 w 0 ) + γ (q 8 x 0 + q 9 y 0 + q 10 z 0 + q 11 w 0 ) + δ (q 1 x 0 + q 13 y 0 + q 14 z 0 + q 15 w 0 )) r j α j(x 0 + q 1 y 0 + q z 0 + q 3 w 0 ) r j = 0, q 4 = 0. Ths s dentcally zero for all (x 0 : y 0 : z 0 : w 0 ) precsely n the ntersecton of the proper transforms of all pont condtons and the exceptonal dvsor. Clearly ths happens when a factor of the frst product s zero for each of the ponts (1 : 0 : 0 : 0), (0 : 1 : 0 : 0), (0 : 0 : 1 : 0), (0 : 0 : 0 : 1). So we get that ths ntersecton s the unon (over ) of the followng sets q 4 = 0 β + γ q 8 + δ q 1 = 0 β q 5 + γ q 9 + δ q 13 = 0 β q 6 + γ q 10 + δ q 14 = 0 β q 7 + γ q 11 + δ q 15 = 0 Each of these, however, we recognze as the ntersecton of the proper transforms of a 14

22 dstngushed P 11 contanng the blow-up center P 3 wth the exceptonal dvsor. Indeed, a P 11 contanng the P 3 s the P 11 whch corresponds to a plane from the frst product ( ) n the equaton of S. So t conssts of all φ s for whch β (φ 4 x 0 + φ 5 y 0 + φ 6 z 0 + φ 7 w 0 ) + γ (φ 8 x 0 + φ 9 y 0 + φ 10 z 0 + φ 11 w 0 ) + δ (φ 1 x 0 + φ 13 y 0 + φ 14 z 0 + φ 15 w 0 ) 0,.e., t s the set n A 15 (p 1, p,..., p 15 ) gven by β p 4 + γ p 8 + δ p 1 = 0 β p 5 + γ p 9 + δ p 13 = 0 β p 6 + γ p 10 + δ p 14 = 0 β p 7 + γ p 11 + δ p 15 = 0 The ntersecton of the proper transform of ths wth the exceptonal dvsor s clearly the clamed component of the unon above. Thus the ntersecton of the proper transforms of the pont condtons n each of the dsjont exceptonal dvsors n V s the unon of the proper transforms of the dstngushed P 11 s contanng the blow-up center yeldng ths exceptonal dvsor. Ths mples (). Theorem.5. Let V be the varety obtaned from V after all blow-ups centered at the proper transforms of the dstngushed P 7 s under the frst set of blow-ups from Theorem.5.1 Then () The proper transforms of the dstngushed P 11 s n V are dsjont. () The multplcty of a pont condton n V along the proper transform of each P 7 correspondng to a lne of the confguraton S equals the multplcty of S along ths lne. () The ntersecton of the proper transforms of the pont condtons n V s the dsjont unon of the proper transforms of the dstngushed P 11 s. Proof Agan, () follows drectly from (), () n Theorem.5.1. As n Theorem.5.1, t s enough to consder only one of the blow-up centers n ths seres of blowng-ups as they are all dsjont. Consder a dstngushed P 7 contanng a dstngushed P 3. Wthout loss of generalty, the P 3 s the same as n Theorem.5.1 and the P 7 corresponds to the lne z = w = 0. In the affne chart ths s {(p 1, p,..., p 15 ) A 15 /p 8 = p 9 = = p 15 = 0}. After blowng-up along the P 3 (as n Theorem.5.1), the proper transform of the P 7 has affne equatons q 8 = q 9 = = q 15 = 0 and the exceptonal dvsor s q 4 = 0. If we next blow-up along ths proper transform of P 7, then the map between a chart A 15 (s 1, s,..., s 15 ) n the blow-up and A 15 (q 1, q,..., q 15 ) s gven by q = s, = ; q 9 = s 8 s 9, q 10 = s 8 s 10,..., q 15 = s 8 s 15 where s 8 = 0 s the exceptonal dvsor. 15

23 The proper transform of the pont condton from Theorem.5.1 then pulls back to the new blow-up as (β (x 0 + s 5 y 0 + s 6 z 0 + s 7 w 0 ) + γ s 8 (x 0 + s 9 y 0 + s 10 z 0 + s 11 w 0 ) + δ s 8 (s 1 x 0 + s 13 y 0 + s 14 z 0 + s 15 w 0 )) r j (α j(x 0 + s 1 y 0 + s z 0 + s 3 w 0 ) + β j s 4 (x 0 + s 5 y 0 + s 6 z 0 + s 7 w 0 ) + γ j s 4 s 8 (x 0 + s 9 y 0 + s 10 z 0 + s 11 w 0 ) + δ j s 4 s 8 (s 1 x 0 + s 13 y 0 + s 14 z 0 + s 15 w 0 )) r j = 0 where α j 0. Thus s 8 factors out from factors n the frst product ( ) only and these are the factors for whch β = 0. Hence the multplcty of the proper transform of the P 7 n the proper transform of a pont condton s r where r s the multplcty of a plane wth β = 0,.e., a plane contanng the lne z = w = 0 correspondng to the P 7. Ths proves () on an open set of the proper transform of P 7 (because we dd not take nto account all other P 3 s n the P 7 ), hence the multplcty n () s the multplcty just found. From the pull-back of a pont condton above we fnd ts proper transform to be β 0 (β (x 0 +s 5 y 0 +s 6 z 0 +s 7 w 0 )+γ s 8 (x 0 +s 9 y 0 +s 10 z 0 +s 11 w 0 )+δ s 8 (s 1 x 0 +s 13 y 0 +s 14 z 0 + s 15 w 0 )) r β =0 (γ (x 0 + s 9 y 0 + s 10 z 0 + s 11 w 0 ) + δ (s 1 x 0 + s 13 y 0 + s 14 z 0 + s 15 w 0 )) r j (α j(x 0 + s 1 y 0 + s z 0 + s 3 w 0 ) + β j s 4 (x 0 + s 5 y 0 + s 6 z 0 + s 7 w 0 ) + γ j s 4 s 8 (x 0 + s 9 y 0 + s 10 z 0 + s 11 w 0 ) + δ j s 4 s 8 (s 1 x 0 + s 13 y 0 + s 14 z 0 + s 15 w 0 )) r j = 0. The ntersecton of ths and the exceptonal dvsor s 8 = 0 then s gven by β 0 (β (x 0 +s 5 y 0 +s 6 z 0 +s 7 w 0 )) r β =0 (γ (x 0 +s 9 y 0 +s 10 z 0 +s 11 w 0 )+δ (s 1 x 0 +s 13 y 0 + s 14 z 0 + s 15 w 0 )) r j (α j(x 0 + s 1 y 0 + s z 0 + s 3 w 0 ) + β j s 4 (x 0 + s 5 y 0 + s 6 z 0 + s 7 w 0 )) r j = 0, s 8 = 0. So the ntersecton of the proper transforms of all pont condtons and the exceptonal dvsor happens where the above s dentcally zero for all ponts (x 0 : y 0 : z 0 : w 0 ). Thus ths ntersecton s the unon (over and j) of the followng sets γ + δ s 1 = 0 α j + β j s 4 = 0 γ s 9 + δ s 13 = 0 α j s 1 + β j s 4 s 5 = 0 γ s 10 + δ s 14 = 0, α j s + β j s 4 s 6 = 0 γ s 11 + δ s 15 = 0 α j s 3 + β j s 4 s 7 = 0 s 8 = 0 s 8 = 0 Now (as n Theorem.5.1) we recognze the frst set as the proper transform of a P 11 correspondng to a plane contanng the lne assocated wth the P 7 we consder. The second set s the proper transform of a P 11 whose plane contans nether the lne nor the pont correspondng to the dstngushed P 7 and P 3. Because the proper transforms of all other P 11 s are dsjont from the exceptonal dvsor s 8 = 0, we see that the ntersecton of the proper transforms of all pont condtons and the exceptonal dvsor conssts of the proper transforms of all P 11 s whch ntersect the exceptonal dvsor. An analogous statement s 16

24 true for the rest of the dsjont blow-up centers. Agan, ths proves () on an open set of the ntersecton n V, but the equatons of the ntersecton components we found ndcate that none of them are contaned n the dvsor s 4 = s 8 = 0 of the dvsor s 8 = 0 (because α j 0), so () holds on the closure of the complement of the dvsors n the exceptonal dvsor we get from the prevous set of blowups. Theorem.5.3 Let V = Ṽ be the varety obtaned from V after all blow-ups centered at the proper transforms of the dstngushed P 11 s under the frst and second sets of blow-ups from Theorem.5.1 and Theorem.5.. Then () The ntersecton of the proper transforms of the pont condtons n Ṽ s empty. () The multplcty of a pont condton n V along the proper transform of each P 11 correspondng to a plane of the confguraton S equals the multplcty of S along ths plane. Proof Anytme we need to consder a P 11 contanng a P 7 contanng a P 3, we can safely assume that the P 3 and P 7 are as n the prevous theorems and the P 11 n the affne chart s gven by p 1 = = p 15 = 0,.e., correspondng to the plane w = 0 (see prevous secton). In the two affne charts of the frst two blow-ups from Theorems.5.1 and.5. the proper transforms of ths P 11 are clearly q 1 = = q 15 = 0 and s 1 = = s 15 = 0. Now the blow-up centered at the latter restrcted on the affne set A 15 (t 1, t,..., t 15 ) s gven by the equatons s = t, = ; s 13 = t 1 t 13, s 14 = t 1 t 14, s 15 = t 1 t 15 where t 1 = 0 s the exceptonal dvsor. The proper transform of a pont condton n V (from Theorem.5.) pulls-back to β 0 (β (x 0 + t 5 y 0 + t 6 z 0 + t 7 w 0 ) + γ t 8 (x 0 + t 9 y 0 + t 10 z 0 + t 11 w 0 ) + δ t 8 t 1 (x 0 + t 13 y 0 + t 14 z 0 + t 15 w 0 )) r β =0 (γ (x 0 + t 9 y 0 + t 10 z 0 + t 11 w 0 ) + δ t 1 (x 0 + t 13 y 0 + t 14 z 0 + t 15 w 0 )) r j (α j(x 0 + t 1 y 0 +t z 0 +t 3 w 0 )+β j t 4 (x 0 +t 5 y 0 +t 6 z 0 +t 7 w 0 )+γ j t 4 t 8 (x 0 +t 9 y 0 +t 10 z 0 +t 11 w 0 )+δ j t 4 t 8 t 1 (x 0 + t 13 y 0 + t 14 z 0 + t 15 w 0 )) r j = 0. The ntersecton of the proper transform of ths and the exceptonal dvsor t 1 = 0 then s β 0 (β (x 0 + t 5 y 0 + t 6 z 0 + t 7 w 0 ) + γ t 8 (x 0 + t 9 y 0 + t 10 z 0 + t 11 w 0 )) r β =0,γ =0,δ 0 (δ (x 0 + t 13 y 0 + t 14 z 0 + t 15 w 0 )) r β =0,γ 0 (γ (x 0 + t 9 y 0 + t 10 z 0 + t 11 w 0 )) r j (α j(x 0 + t 1 y 0 + t z 0 + t 3 w 0 ) + β j t 4 (x 0 + t 5 y 0 + t 6 z 0 + t 7 w 0 ) + γ j t 4 t 8 (x 0 + t 9 y 0 + t 10 z 0 + t 11 w 0 )) r j = 0, t 1 = 0. Agan, we need to see where ths s dentcally zero n the exceptonal dvsor. From the frst and the last product we fnd that ths s the unon of the followng sets: 17

25 β + γ t 8 = 0 α j + β j t 4 + γ j t 4 t 8 = 0 β t 5 + γ t 8 t 9 = 0 α j t 1 + β j t 4 t 5 + γ j t 4 t 8 t 9 = 0 β t 6 + γ t 8 t 10 = 0, α j t + β j t 4 t 6 + γ j t 4 t 8 t 10 = 0 β t 7 + γ t 8 t 11 = 0 α j t 3 + β j t 4 t 7 + γ j t 4 t 8 t 11 = 0 t 1 = 0 t 1 = 0 Followng the equatons of all blow-ups, we see that these are precsely the ntersectons of the proper transforms of the P 11 s wth the exceptonal dvsor. However, accordng to () n Theorem.5., these ntersectons are empty. Ths proves () on an open set of the blow-up but the above equatons show that the ntersecton s empty n t 4 = t 8 = t 1 = 0 (because β 0, α j 0) whch takes care of all closed sets n the blow-up we dd not consder (correspondng to all P 3 s and P 7 s n the P 11 ). Hence () s verfed. To show (), we could stll use the dea of () n Theorem.5. and the proper transform of a pont condton found n (). In fact, we can smplfy everythng (and make t more lke () of Theorem.5.1) by consderng the bratonal mages n V of all closed sets nvolved. Thus we need to fnd the multplcty of a dstngushed P 11 n a pont condton n V. Wthout loss of generalty, a P 11 s gven by p 1 = = p 15 = 0 n the affne pece of V,.e., t corresponds to the plane w = 0. Then the blow-up centered at ths P 11 s gven by p = q, = ; p 13 = q 1 q 13, p 14 = q 1 q 14, p 15 = q 1 q 15 where q 1 = 0 s the exceptonal dvsor n a chart A 15 (q 1, q,..., q 15 ) of the blow-up. Usng these equatons, we blow-up a pont condton n V and get (β (q 4 x 0 + q 5 y 0 + q 6 z 0 + q 7 w 0 ) + γ (q 8 x 0 + q 9 y 0 + q 10 z 0 + q 11 w 0 ) + δ q 1 (x 0 + q 13 y 0 + q 14 z 0 + q 15 w 0 )) r j (α j(x 0 + q 1 y 0 +q z 0 +q 3 w 0 )+β j (q 4 x 0 +q 5 y 0 +q 6 z 0 +q 7 w 0 )+γ j (q 8 x 0 +q 9 y 0 +q 10 z 0 +q 11 w 0 )+δ j q 1 (x 0 + q 13 y 0 + q 14 z 0 + q 15 w 0 )) r j = 0. Snce α j 0, q 1 can be factored out from the frst product only and ths s possble when β = γ = 0 for some. Hence the multplcty of the exceptonal dvsor n the pull-back of the pont condton s r, the multplcty of the plane w = 0 (δ w = 0) and () s proved. Let π be the composton of all blowng-ups from the theorems,.e., Ṽ V V V = P 15 Recall that the ratonal map s s regular on an open set U, the complement of the base locus n V. The nverse mage Ũ of U n Ṽ s somorphc to U and the restrcton of π onto Ũ nduces a regular map s from Ũ to PN (when ratonal map from Ṽ to PN for whch s π = s. composed wth s). Thus s := s π s a Then the lnear system of s conssts of the pull-backs to Ṽ of the elements of the lnear 18

26 system of s. All these pull-backs have common components along the exceptonal dvsors of the blow-ups,.e., nsde the complement of Ũ. So f we cancel these components out, we wll (re)defne the same s (wth a new lnear system) and t wll stll form a commutatve dagram (see below) wth π and s as ratonal maps. Ths, of course, can be acheved by consderng the proper transforms of the elements of the lnear system of s nstead of ther nverse mages because both agree on Ũ and the proper transforms do not have components along the exceptonal dvsors (they are closures of ther restrctons to Ũ). In secton.3 we saw (as a consequence of Proposton.3.1) that the lnear system of s s spanned by the pont condtons n V and n ths secton we proved that the proper transforms of the latter have an empty ntersecton n Ṽ. Hence the ratonal map s s n fact regular because ts lnear system s spanned by the proper transforms of the pont condtons. As we noted above, t s compatble wth π and s on followng Theorem.5.4 The varety Ṽ commutatve dagram: Ũ, so all these arguments prove the resolves the ndetermnaces of s,.e., there s a wth s a regular map and O S = Im( s). π Ṽ V s s P N The multplcty statements n the theorems wll be used later n chapter three..6 Proof of Geometrc Interpretaton To prove Theorem..1, we wll need a couple of lemmas frst. Usng the expressons from secton.5, we see that f S s a reduced plane confguraton, then a pont condton n V correspondng to a pont (x : y : z : w) s j (α jxφ 0 + α j yφ 1 + α j zφ + α j wφ 3 + β j xφ 4 + β j yφ 5 + β j zφ 6 + β j wφ 7 + γ j xφ 8 + γ j yφ 9 + γ j zφ 10 + γ j wφ 11 + δ j xφ 1 + δ j yφ 13 + δ j zφ 14 + δ j wφ 15 ) = 0 where α j x + β j y + γ j z + δ j w = 0, j 1 are the planes n S. In partcular, the pont condtons n P 15 are hyperplane confguratons and t wll be convenent to thnk of these hyperplanes as of ponts of ˇP 15, the coordnates beng ther coeffcents,.e., the hyperplanes 19

27 of the above pont condton are the ponts (α j x : α j y : α j z : α j w : β j x : β j y : β j z : β j w : γ j x : γ j y : γ j z : γ j w : δ j x : δ j y : δ j z : δ j w). Lemma.6.1 For every nonempty set X n P N, there s an open set U n P 3, such that X s contaned n none of the pont condtons n P N correspondng to the ponts of U. Proof Gven X P N, consder a pont x n t. It corresponds to a (degree d) surface S x n P 3. Let U be the complement of S x n P 3. Let P be a pont condton n P N whch corresponds to a pont p n U. It does not contan X because t does not contan x. Indeed, otherwse S x would contan p,.e., S x would have a nonempty ntersecton wth ts complement U. The contradcton shows that the open set U we constructed satsfes the requrements of the lemma. Lemma.6. A hyperplane of a pont condton n P 15 does not contan an ntersecton of hyperplanes of any other pont condton. Proof Suppose that (the support of) a hyperplane component (αx : αy : αz : αw : βx : βy : βz : βw : γx : γy : γz : γw : δx : δy : δz : δw) of the pont condton for (x : y : z : w) contans the ntersecton of the hyperplane components (α j x : α j y : α j z : α j w : β j x : β j y : β j z : β j w : γ j x : γ j y : γ j z : γ j w : δ j x : δ j y : δ j z : δ j w ) of another pont condton correspondng to the pont (x : y : z : w ). Ths s possble f and only f the frst vector s a nontrval lnear combnaton of the others. Wthout loss of generalty α 0. Then at least one of the α j s s nonzero too (x, y, z, w are not all zero). Now, restrctng the lnear combnaton to the frst four coordnates only, we have that (αx : αy : αz : αw) s a lnear combnaton of the vectors (α j x : α j y : α j z : α j w ),.e., (x : y : z : w) s a lnear combnaton of α j (x : y : z : w ) whch mples that ether x = y = z = w = 0 or (x : y : z : w) = (x : y : z : w ), a contradcton to our assumpton n ether case. Corollary.6.3 Each rreducble component of the ntersecton of any k pont condtons n P 15 s the ntersecton of at most k hyperplane components (at most one from each pont condton). We also need G, the blow-up of V = P 15 along the subscheme defned by the deal generated by the components (coordnate equatons) of the ratonal map s : P 15 P N (ths subscheme s supported on the base locus of s). The varety G s somorphc to the closure of the graph of s n the product P 15 P N. 0

28 G also resolves the ndetermnaces of s, lke Ṽ from the prevous secton dd, and n fact G s the unversal object among all such blow-ups (pg.164, [6]). Except for ths secton however, Ṽ wll be much more convenent for our purposes. For G and the regular maps (projectons) to P 15 and P N we stll have a commutatve dagram. G π π 1 P 15 s P N Wth all ths n mnd we are ready to prove Theorem.6.4 If the dmenson of the orbt of a reduced plane confguraton S s j, then the degree of the orbt closure equals the number of translates of S whch pass through j general ponts. Proof Frst we show that there are j general pont condtons n P N (correspondng to j general ponts n P 3 ) whose ntersecton wth the orbt closure O S s zero dmensonal. Indeed, every pont condton, beng a hyperplane, ntersects O S (j > 0) and accordng to lemma.6.1, there s a dense open set U 1 n P 3 for whch the correspondng ntersectons are of dmenson j 1. Next we apply the same argument for each of the rreducble components of ths ntersecton: for each of those components there s a dense open set of P 3 whose correspondng pont condtons ntersect t n a j dmensonal varety. So for the ntersecton U of all these open sets we have that the ntersecton of any pont condton assocated to a pont of U wth the frst pont condton correspondng to a pont of U 1 and O S s of dmenson j. We terate ths process tll we get j open sets U, = 1,..., j n P 3 respectvely parameterzng j pont condtons P, = 1,..., j for whch ( P ) O S s zero dmensonal. Next we prove that the ponts of ntersecton found above can be n the orbt f the choce of pont condtons s stll general. The tools (Lemma.6.1) and the dea are the same. O S O S s n the boundary of O S whose dmenson s at most j 1. So when we choose the frst j 1 pont condtons we can further restrct the open sets U (by ntersectng them wth other dense open sets) so that the process of gettng strctly decreasng by one dmensons of the ntersectons (after ntersectng wth each pont condton) holds for the boundary of the orbt closure as well as the orbt closure. Fnally the last j-th pont condton can be chosen so as to avod the zero-dmensonal ntersecton of the frst j 1 pont condtons wth the 1