OVALS OF THE REAL PLANE OF ALGEBRAIC CURVES, OF INVOLUTIONS OF FOUR-DI - PDF document

OVALS OF THE REAL PLANE OF ALGEBRAIC CURVES, OF INVOLUTIONS OF FOUR-DIMENSIONAL SMOOTH MANIFOLDS, THE ARITHMETIC OF INTEGER-VALUED QUADRATIC FORMS on the certain four-dimensional s2. Structure of Four-Dimensional Manifold = 0 yare real variables Since the (x0 : z 2 -= F x 2 algebraic surface : E ~ ~ = : x~)}), by the z 2 = two-sheeted ramified clear from our algebraic F = real dimension and Form - 1 our complex such that We denote by H2(Y) = two-dimensional integer-valued ( , ) bilinear integer-valued y ~ Y ) ~ bilinear form ~r b E Form~vis symmetric = + on Y, §4. Arithmetic Lemma • : Z r × Z r integer-valued symmetric • = + bilinear form. LEMMA 2. w E Z r x E Z r x) rood rood 8 for where, according Computing the Fundamental Class r : ~ y4S y4S 2s] repre- sented by cycle A '2s. lying on A ~ Y index of of ([A], [A]) = = 2k 2 the index But in A) = Real Part values of : x2. F, it is note that of negative to the coordinate axes and then Euler characteristic ovals by Euler characteristic ~7. Homologies between the Cycles of classes represented [A] = [HI ~ x 0 = 0 x 0 : x i : x 2 shall assume that this n = or from from the homology class of cycle ~o in H2(CP2). Each cycle c of H2(CI~) is homologous to (c, co) loci. For example, example, its homology class in H2(Y). Lemma 9 follows from the two relationships [AI = k k (: Ha (r), (7) [H] = k [ ~ prove relationship 1)A - 2koo teger-vahed three-dimensional K 3 K~ = = k[ooy] is a second-order element is the we note (this is Harnack's Theorem). compact oriented F = 0 B ÷ boundaries are F = 0 was defined combinatory, generally compact two-dimensional ~. Therefore, surface B + C that the index B ÷ C B • x = that its x = £ � 0 x = transversally intersects of along an x = 0 with line x = of PA (because this earlier assertion assertion +C] = k[~] EH 2 (CP 2, Z2). There- fore, the cycle B + a Z 2 Z 2 L 3 in CP = B + C C (Z2). Lemma 9 is proven. §8. Proof of Theorem w = [A] E E E H2(Y). According to Lemma 1, the form is symmetric and nonsingular; according to Lemma 3, class w is fundamental. By Lemma 9, class w' differs from w on even elements. Thus, Lemma 2 is applicable, and we find from Eq. (6) that ~D~ ([A], [A]) ~ (IHI, (II) = W, W, [A]) = = lI]) =-(l-I II). Therefore, (A, A) + (fl, II) - 0 (II, n) = 0 mod §9. Remarks of D. was studied studied by means of the Hirzebruch-Atiyah- Zinger signature formulas ([10], ~ 6). By joining these computations with ours and with the Lefschetz- Dold-Atiyah-Bott formulas given by Hirzebruch in [11], we can obtain additional information on manifold Y and its Y ~ Y of the H = Z 2 + Z 2 those points in x 0 : x 1 : x 2 F = 0 Fundamental class of of k [oor] k [o%] Trace of involution ~h 2 2 2 n t (k ÷ these results were but without Tobe more can also real curves. For from relationships Mcurves); [l÷Tr~.l~3~--3k÷l. forms ~a ~ r same parity forms ~l our assertion. ~r are k = = Chapter 5j therefore permit r, ~ar k = 2 of degree = ~ r r the form F 8 is of simple proof of I. G. E = scalar pro- duct given by scalar products, direct sum proper subspaces E i operator a.. degenerate. Therefore, E 1 = g~ + used here 5 = 5 = =a÷b--c--d, Trl.=a+b÷c÷d, = a c @ • l = a signatures from from ÷ d, which also proves inequality (2). 5. Our constructions also lead to new constraints on the distribution of ovals. In order to formulate these constraints, we partition all ovals into three classes as a function of the sign of the Euler character- is an Euler characteristic m 0, m_, so that 2) , /n_~ (k-- p+~b, m+~d, 2 2 where numbers Ti. Of Of E H~(Y) and the the assumes positive values. More- over, ~, [Ili] = [Hi]. It follows from this that that are linearly independent, and that on the plane L span- ning them, form ¢1 is L ~ E + = = 2 ' - 2 ' - k k 2 + = k 7. We X = = ~ (Sgn Y k ~ = a~d, c~b, ~ 0 (roodS). these relationships which is without which of A. the joint thank A. LITERATURE CITED und bei Atiyah and of elliptic 23, No.