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Does there exist an irreducible curve of degree 11 in the projective plane which would have 15 triple points?

For information, such a curve would be rational, if it exists, and would be smooth at all other points (compute the genus with the classical formula: $(d-1)(d-2)/2-\sum a_i(a_i-1)/2=45-3\cdot 15=0$)

flag	asked May 30 at 23:43 Jérémy Blanc 1,598●1●4●13

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if we start from three lines with a common point, then add two lines through some point of that triad, we have degree 5 and only two triple points, so adding 6 more lines gives only 14 triple points. thus even if we can smooth the extraneous double points we lose. maybe one should pass some conics through more nodes. we could pass one conic through all 4 nodes at step 3 above, getting a degree 7 curve with 6 triple points, and 4 nodes. passing another conic through all 4 gives a degree 9 curve with 10 triple points, and 10 nodes. so we get a reducible one, then try to smooth it. / – roy smith May 31 at 2:24

Igor Rivin · Answer 1 · 2012-05-30 23:52:47Z UTC

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This seems to be study in this paper by O. Dumitrescu (which I did not read in detail, but it seems quite algorithmic).

answered May 30 at 23:52

Igor Rivin
31.2k●3●37●71

According to the paper, the virtual dimension of the linear system is $78-90-1<0$ since the degree of the equation is at least $5$. So it looks like the answer is no. – Will Sawin May 31 at 3:00

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Dumitrescu's paper deals with points in general position (even if that may not be clear in the intro). The singular points of that rational curve would certainly not be in general position; that can happen. For instance, there are indeed curves of degree 10 with 12 double points. These two papers by Gradolato-Mezzetti are more relevant: "Families of curves with ordinary singular points on regular surfaces". Ann. Mat. Pura Appl. (4) 150 (1988), 281–298. and "Curves with nodes, cusps and ordinary triple points", Ann. Univ. Ferrara Sez. VII (N.S.) 31 (1985), 23–47 (1986). It's Roy's approach. – quim May 31 at 15:22

I ment 12 triple points! – quim Jun 1 at 6:03

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Yes, the points could be in special position, of course. The simplest example is sextic with 10 double points. A lot of such curves exist, but the points are not in general position. I will try to find the articles you mention. They seem to be very interesting. – Jérémy Blanc Jun 13 at 16:55

I found the articles and some rough bounds which give existence of curves with triple points, but the bounds seem to be too far from my explicit question. – Jérémy Blanc Jun 25 at 14:57

Does there exist a curve of degree 11 having 15 triple points?

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