Abstract Generated abstract
The paper studies when selected ordinary double points of a real plane algebraic curve can be varied by small perturbations of the curve while preserving the total configuration of simple singularities. It formulates variability in the parameter space of curves of fixed degree and relates it to linear independence of the conditions imposed by prescribed real and conjugate imaginary double points. For irreducible curves, the argument uses linear series cut out by curves of the same degree and derives necessary dimension inequalities, with complete criteria in genus zero and sufficient or obstructing conditions in positive genus. The results generalize earlier work on real double points and include nonvariability in the case where the imposed real double point conditions exhaust the parameter dimension.
Full Text
D. A. Gudkov
VARIABILITY OF SIMPLE DOUBLE POINTS OF REAL PLANE ALGEBRAIC CURVES*
(Presented by Academician I. G. Petrovskii, 20 X 1961)
- We shall adopt the following notation and terminology: \((x:y:z)\) are the coordinates of a point in the complex projective plane \({}^{*}R^{2}\); \(R^{2}\) is the real part of \({}^{*}R^{2}\); \(E^{2}\) is the complex affine plane obtained from \({}^{*}R^{2}\) by putting \(z=1\); \(E^{2}\) is the real part of \({}^{*}E^{2}\). Suppose that a unitary metric is fixed in \({}^{*}E^{2}\), and consequently a Euclidean metric in \(E^{2}\). \(\rho^{*}(a^{0})\) denotes the \(\rho\)-neighborhood of the point \(a^{0}\in{}^{*}E^{2}\) in \({}^{*}E^{2}\); \(\rho(a^{0})\), the \(\rho\)-neighborhood of the point \(a^{0}\in E^{2}\) in \(E^{2}\).
Let, by the equation
\[ H(x,y,z)\equiv \sum_{\alpha+\beta+\gamma=m} A_{\alpha\beta\gamma}x^{\alpha}y^{\beta}z^{\gamma}=0 \tag{1} \]
there be given a real algebraic curve of order \(m\) in the plane \({}^{*}R^{2}\). Such a curve corresponds one-to-one to a “point” of the real projective space \(R^{N}\) \((A_{00}:A_{10}:A_{01}:\ldots:A_{0m})\) of dimension
\[ N=\frac{m(m+3)}{2} \]
\(({}^{2}), p. 75). The terms “point” and “curve” in \(R^{N}\) will be put in quotation marks, in distinction from the same terms in \({}^{*}R^{2}\). The curve \(H\) will also be called the “point” \(H\) in \(R^{N}\). Let \(E^{N}_{\alpha_{0}\beta_{0}}\) denote the affine space obtained from \(R^{N}\) by setting the coefficient \(A_{\alpha_{0}\beta_{0}}\) equal to one. We regard \(E^{N}_{\alpha_{0}\beta_{0}}\) as a Euclidean space. \(S(F,\varepsilon)\) denotes the \(\varepsilon\)-neighborhood of the “point” \(F\in E^{N}_{\alpha_{0}\beta_{0}}\) in \(E^{N}_{\alpha_{0}\beta_{0}}\). A curve \(F\) having \(\delta\) simple double points and having no other singular points in \({}^{*}R^{2}\) will be called a \(\delta\)-simple curve (or, more briefly, a simple curve).
- Definition of the variability of simple double points. Let the curve
\[ F\equiv \sum_{0\leq \alpha+\beta\leq m} A^{0}_{\alpha\beta}x^{\alpha}y^{\beta}=0 \]
be a \(\delta\)-simple curve, all the singular points of which belong to \({}^{*}E^{2}\). Let \(b^{0}_{j}(x^{0}_{j},y^{0}_{j})\) \((j=1,2,\ldots,(k+q))\) be real simple double points, and let \(a^{0}_{\nu}(x^{0}_{\nu},y^{0}_{\nu})\) and \(\overline{a}^{0}_{\nu}\) \((\nu=1,2,\ldots,(l+s))\) be pairs of imaginary conjugate simple double points of the curve \(F\) \((k+q+2(l+1)=\delta)\). We shall call \(k\) real points \(b^{0}_{j}\) \((j=1,2,\ldots,k)\) and \(l\) pairs of imaginary conjugate points \(a^{0}_{\nu}\) and \(\overline{a}^{0}_{\nu}\) \((\nu=1,2,\ldots,l)\) \(v(\varepsilon,\rho)\)-variable (or, more briefly, variable), if, for some coefficient \(A_{\alpha_{0}\beta_{0}}\) \((A^{0}_{\alpha_{0}\beta_{0}}\neq 0)\), for any sufficiently small \(\varepsilon>0\) and \(\rho>0\) one can find such a \(v(\varepsilon,\rho)>0\) that, choosing arbitrarily points \(b_{j}\in v(b^{0}_{j})\) \((j=1,2,\ldots,k)\), \(a_{\nu}\in v^{*}(a^{0}_{\nu})\) and \(\overline{a}_{\nu}\in v^{*}(\overline{a}^{0}_{\nu})\) \((\nu=1,2,\ldots,l)\), one can find a curve
\[ G=\sum_{0\leq \alpha+\beta\leq m} \widetilde{A}_{\alpha\beta}x^{\alpha}y^{\beta}=0 \]
* We have already considered this question in \(({}^{3})\).
and such a continuous “curve” \(FG \subset S(F,\varepsilon) \subset E_{\delta_0\beta_0}^{\gamma}\), that:
1) every curve \(H \in FG\) is a \(\delta\)-simple curve;
2) the ordinary double points of the curve \(H\) are located one in each of the neighborhoods \(v(b_j^0)\) \((j=1,2,\ldots,k)\), \(v^*(a_\nu^0)\) and \(v^*(\overline{a_\nu^0})\) \((\nu=1,2,\ldots,l)\), \(\rho(b_j^0)\) \((j=(k+1),\ldots,(k+q))\), \(\rho^*(a_\nu^0)\) and \(\rho^*(\overline{a_\nu^0})\) \((\nu=(l+1),\ldots,(l+s))\) (pairwise nonintersecting);
3) the coordinates of the singular points of the curve \(H\) are continuous functions of the coefficients of the curve \(H\), if \(H \in FG\);
4) the curve \(G\) has singular points at the chosen points \(b_j\) \((j=1,2,\ldots,k)\), \(a_\nu\) and \(\overline{a_\nu}\) \((\nu=1,2,\ldots,l)\), and one singular point in the \(\rho^*\)-neighborhood of each of the remaining singular points of the curve \(F\)*.
Lemma 1 . If the system of \((3k+6l+q+2s)\) linear equations with respect to all coefficients \(A_{\alpha\beta}\) of the curve \(H\)
\[ H(b_j^0)=0,\qquad H_x(b_j^0)=0,\qquad H_y(b_j^0)=0 \quad (j=1,2,\ldots,k); \tag{2^1} \]
\[ H(a_\nu^0)=0,\qquad H_x(a_\nu^0)=0,\qquad H_y(a_\nu^0)=0, \]
\[ H(\overline{a_\nu^0})=0,\qquad H_x(\overline{a_\nu^0})=0,\qquad H_y(\overline{a_\nu^0})=0 \quad (\nu=1,2,\ldots,l); \tag{2^2} \]
\[ H(b_j^0)=0 \quad (j=(k+1),\ldots,(k+q)); \tag{2^3} \]
\[ H(a_\nu^0)=0,\qquad H(\overline{a_\nu^0})=0 \quad (\nu=(l+1),\ldots,(l+s)) \tag{2^4} \]
is linearly independent, then the \(k\) points \(b_j^{(0)}\) \((j=1,2,\ldots,k)\) and the \(2l\) points \(a_j^0\) and \(\overline{a_\nu^0}\) \((\nu=1,2,\ldots,l)\) of the curve \(F\) are variable.
Theorem 1. For the variability of \(k\) real and \(l\) pairs of imaginary conjugate ordinary double points of the curve \(F\), it is necessary that the inequality
\[ 3(k+2l)+(q+2s)\leq N. \tag{3} \]
Lemma 2. Let the curve \(F\) be irreducible, \(l=0\), and let the \(k\) ordinary double points \(b_j^0\) \((j=1,2,\ldots,k)\) of the curve \(F\) be variable. Then the indicated \(k\) ordinary real double points of the curve \(F\) can be displaced (arbitrarily little) to such points \(b_j\) \((j=1,2,\ldots,k)\) that the system of equations
\[ H(b_j)=0,\qquad H_x(b_j)=0,\qquad H_y(b_j)=0 \quad (j=1,2,\ldots,k) \tag{4} \]
(with respect to the coefficients \(A_{\alpha\beta}\) of the curve \(H\)) will be linearly independent.
- Let the curve \(F\) be irreducible. Consider the linear series \(g_n^r\), cut out on \(F\) by the linear system of curves of order \(m\) determined by the system of equations (2).
Denote \(k+2l=h\), \(q+2s=g\). We introduce a single notation for all singular points of the curve \(F\). Let \(c_j^0\) \((j=1,2,\ldots,h)\) be the ordinary double points whose variability is under study, and let \(c_j^0\) \((j=(h+1),\ldots,(h+g))\) be the remaining singular points of the curve \(F\). Let \(P_j\) and \(Q_j\) \((j=1,2,\ldots,(h+g))\) be the branches of the curve \(F\) with center at the point \(c_j^0\). Consider the cycles
\[ D=\sum_{j=1}^{h+g}(P_j+Q_j),\qquad B=\sum_{j=1}^{h}(P_j+Q_j). \]
* In (3) we gave the definition of variability for the curve \(F\) having only real ordinary double points.
** Lemma 1 is a generalization of Theorem 7 (3).
It is known that the series \(g_n^r\) is the residual of the series cut out on the curve \(F\) by all curves of order \(m\), with respect to the cycle \(D + B\), and \(g_n^r\) is a complete series \(\left( (^{2}),\ \text{p. }211,\ \text{vol. }6.3 \right)\).
In what follows we shall assume that condition (3) is satisfied (otherwise the question is settled by Theorem 1). Since \(F\) is irreducible, we have
\[ h + g \leqslant \frac{m(m-3)}{2} + 1 . \tag{5} \]
Adding (3) and (5), we obtain
\[ 4h + 2g \leqslant m^2 + 1 . \tag{6} \]
But the series \(g_n^r\) has order \(n = m^2 - (4h + 2g)\); therefore \(n\) can be less than zero only for a curve of genus \(p = 0\).
Theorem 2*. Let the curve \(F\) have genus \(p=0\). Then, for the variability of any \(k\) real and \(l\) pairs of imaginary conjugate ordinary double points of the curve \(F\), it is necessary and sufficient that the inequality
\[ 2k + 4l \leqslant 3m - 1 . \tag{7} \]
hold.
Lemma 3. Let the curve \(F\) have genus \(p \geqslant 1\). Then, for the linear independence of the system (2), it is necessary and sufficient that the series \(g_n^r\) be nonspecial.
Theorem 3. Let the curve \(F\) have genus \(p \geqslant 1\). Then, for the variability of any of its \(k\) real and \(l\) pairs of imaginary conjugate ordinary double points, it is sufficient that inequality (7) hold.
Lemma 4. Let the curve \(F\) have genus \(p \geqslant 1\) and suppose that the condition
\[ 3(k + 2l) + (q + 2s) = N \tag{8} \]
is satisfied.
Then the system (2) is linearly dependent for any choice of \(k\) real and \(l\) pairs of imaginary conjugate ordinary double points of the curve \(F\).
Theorem 4. Let the curve \(F\) have genus \(p \geqslant 1\). Then, if the conditions
\[ l = 0,\qquad 3k = N, \tag{9} \]
are satisfied, the \(k\) real double points of the curve \(F\) are not variable**.
Research Physico-Technical Institute
of Gorky State University
named after N. I. Lobachevsky
Received
13 X 1961
REFERENCES
- K. Rohn, Math. Ann., 73, 177 (1913).
- R. Walker, Algebraic Curves, Moscow, 1952.
- D. A. Gudkov, DAN, 98, No. 3, 337 (1954).
* Theorem 2 is a strengthening and generalization of Theorem 8 from (³).
** The well-known special case is \(m = 6,\ k = 9,\ p = 1\) (¹).