saimas’s notes on things worth learning, plus the apps and electronics projects I build. In English and Japanese.https://saimas.jpen-ushttps://saimas.jp/notes/spotting-projective-space/?lang=enhttps://saimas.jp/notes/spotting-projective-space/?lang=enA 2026 paper uses the new bound to show that if one squashing direction has length more than n, the space must be projective space. In one line: No length can pass n + 1, and only projective space gets above n.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/extremal-ray-length/?lang=enhttps://saimas.jp/notes/extremal-ray-length/?lang=enThe new theorem sharpens a basic fact of the minimal model program. Every squashing direction holds a rational curve with −K·C at most n + 1. In one line: Wherever a proof used Miyaoka and Mori’s constant 2n, the new theorem can put n + 1 instead.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/n-plus-one-is-optimal/?lang=enhttps://saimas.jp/notes/n-plus-one-is-optimal/?lang=enStart the theorem from a straight line in projective space, and its bound comes out exactly 1, so no constant smaller than n + 1 can work. In one line: A bound is the best possible when some example meets it exactly.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/hyperboloid-made-of-lines/?lang=enhttps://saimas.jp/notes/hyperboloid-made-of-lines/?lang=enFeed the numbers of one circle on Kobe Port Tower into the theorem, and it says a straight line passes through every point of the tower. In one line: Once the bound drops below 2, the only curve that fits is one of size 1, a straight line.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/n-plus-one-per-pin/?lang=enhttps://saimas.jp/notes/n-plus-one-per-pin/?lang=enHolding a pinned point in place costs n knobs, and keeping its mark on a wall costs one more. So each pin costs n + 1, and that is the n + 1 in the theorem. In one line: Divide a family’s knobs by n + 1 to get how many pins it can afford. That is the n + 1 in the theorem.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/bubble-at-the-pin/?lang=enhttps://saimas.jp/notes/bubble-at-the-pin/?lang=enTo make a curve break at a pin you choose, mark a point next to the pin, keep the mark on a wall, and slide the mark into the pin. In one line: To force a break at a chosen pin, slide a marked point into it while the mark stays on a wall that misses the pin.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/broken-piece-is-sphere/?lang=enhttps://saimas.jp/notes/broken-piece-is-sphere/?lang=enMori’s bend-and-break says a pinned curve that keeps moving must break, and every piece that comes off is a rational curve, a sphere. In one line: To find a rational curve, pin a curve and keep moving it until it breaks. What comes off is rational.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/pinned-circle-snaps/?lang=enhttps://saimas.jp/notes/pinned-circle-snaps/?lang=enOn the tower’s surface, a circle pinned at two points stretches as you tilt it, then breaks into two straight lines, one through each pin. In one line: Pin a curve before you move it. With no way to slide, it has to break.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/frobenius-keeps-holes/?lang=enhttps://saimas.jp/notes/frobenius-keeps-holes/?lang=enIn characteristic p, the Frobenius map wraps a curve p times without adding holes, so a curve’s guaranteed knobs can grow without limit. In one line: When holes block you, change the numbers. In characteristic p, wrapping adds no holes.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/wrapping-adds-holes/?lang=enhttps://saimas.jp/notes/wrapping-adds-holes/?lang=enWrapping a curve k times multiplies −K·C by k, but over the complex numbers it adds holes just as fast, so the guaranteed knobs only scale by k. In one line: Over the complex numbers, wrapping a curve can’t turn a negative knob estimate positive.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/holes-cost-knobs/?lang=enhttps://saimas.jp/notes/holes-cost-knobs/?lang=enThe paper estimates a curve’s knobs with one formula, −K·C + n(1 − g), and in it every hole costs as many knobs as one pin. In one line: Each hole in a curve costs n knobs, the same as one pin.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/curve-size-is-crossings/?lang=enhttps://saimas.jp/notes/curve-size-is-crossings/?lang=enThe paper measures a curve not by its length but by how many times a flat ruler crosses it, counted so the answer never changes. In one line: Measure a curve by its crossings with a flat ruler, and count the hidden ones too.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/complex-circle-is-sphere/?lang=enhttps://saimas.jp/notes/complex-circle-is-sphere/?lang=enA complex number is two real numbers, so a curve looks like a surface. A circle, with one point at infinity added, becomes a sphere. In one line: A curve with no holes is a sphere in disguise, and those are the rational curves.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/pin-costs-n-knobs/?lang=enhttps://saimas.jp/notes/pin-costs-n-knobs/?lang=enMaking a curve pass through a fixed spot takes one equation per coordinate, so in n-dimensional space each pin costs n knobs. In one line: Each pin costs as many knobs as the space has dimensions.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/a-curve-is-a-point/?lang=enhttps://saimas.jp/notes/a-curve-is-a-point/?lang=enA circle is fixed by three numbers, so all circles together form a 3-dimensional space, and each circle is one point in it. In one line: To count how freely a curve can move, count the numbers that pick it out.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/dimension-is-address-count/?lang=enhttps://saimas.jp/notes/dimension-is-address-count/?lang=enDimension counts the numbers needed to say where a point is, so you can count it even for shapes no one can picture. In one line: Count a shape’s dimension by the numbers a point’s address needs, not by the directions you can see.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/curved-tower-straight-rods/?lang=enhttps://saimas.jp/notes/curved-tower-straight-rods/?lang=enSome curved surfaces are made entirely of straight lines, and a 2025 paper shows how one curve can reveal such lines. In one line: Whether a straight line fits on a surface depends on its equation, not on how curved it looks.Sun, 11 Oct 2026 00:00:00 GMThttps://saimas.jp/notes/resistor-at-1ghz/?lang=enhttps://saimas.jp/notes/resistor-at-1ghz/?lang=enEven the resistor, a very simple part, works differently at high frequency. In one line: A part works like its circuit symbol only at low frequency.Tue, 29 Sep 2026 00:00:00 GMT