Kustin-Miller unprojection, degree maps and post-nuclear relationships of the future

Karim Adiprasito

Dear Diary,

yesterday I heard an amazing talk by Eva Philippe and I will get back to that, as well as related questions I am thinking about with Sergey Avvakumov (who is on the market btw., though I do not expect trouble there cause he is doing great things) and Mark Berezovik (who I got to Jerusalem to finish his masters in exile. Study in exile. Interesting that we do that again).


Anyway, dear diary, it reminded me of something many years in the future.

You see, in 35 years I am going to sit my second grandchild on my lap. The bright one. She was always the bright one. So many questions.

And so, after helping my first, my grandson, build a gauntlet for the mutated squirrel warriors, I will sit her down. Spit the iodine tablet into the bucket and sigh, thinking about the best advice. She is starting to get interested in relationships, and she is asking me.

“Grandpa, I want to ask about pairings”

she begins, and my bioengineered squidheart sinks. Not that one.

“Grandpa, in a standard Gorenstein algebra over a field…”

just what I feared. She sees the hesitation in my eyes. I am not ready for this, but her parents are not around anymore.

“the identification of the fundamental class A^{\mathrm{top}} with the ground field k… it is so arbitrary. What do I do?” She asks, her eyes bright with childish innocence. I am translating here from the Russo-portuguese creole of the future, of course.

Sigh. Time to have ‘the talk’, apparently. Difficult, but necessary. She needs to be prepared.

“You see, in my time” and I start to explain something that really noone knew at my time, but people should have. Maybe it would have prevented the Rhino riots. I start again. The anthrax dust is making breathing hard.

“When I was young, you see, we realized this.” Both of my arms amputated after the rock-people revolt of 42, I use my incredibly dexterous feet to draw exact sequences into the poisonous grime.

“You see, it is a little bit like with birds and bees. Girls and boys. Like mommy and daddy when they made you and your brother”. My eyes dart to the side, seeing her brother vaporize a militant rodent with the atomizer while another critter stabs a venomous dart into his thigh. Oh what I would give to be out there.

“See, sometimes, a Gorenstein ring R comes together with a codimension one ideal J \subset R.”

“So that R/J is Gorenstein, Grosspapa?”

she asks. Too bright for this world.

“Yes, lambkin” I pet her head, worrying about her future. “And what do we do then?” I ask, half hoping she does not know the answer.

“Dualizing” she blurts out, her eyes a bright green shimmer like mine. Shining like the plutonium fields. My granddaughter.

“That is right, we get:”

0 \rightarrow R \rightarrow \mathrm{Hom}_R(J,R) \rightarrow R/J \rightarrow 0

where the last map corresponds to the Poincar\’e residue map in complex geometry.

“Wow. So there is a \phi \in \mathrm{Hom}_R(J,R) that generates \mathrm{Hom}_R(J,R) as an R-module she blurts out.

“Together with…” I start to correct but she continues my thoughts.

“Together with the inclusion i: J \rightarrow R.”

That is right. I don’t have the heart to tell her that \phi is almost unique. Because she knows. We both know.

I wish I could spare her this future. But I continue.

“And then Reid defined the unprojection

\mathrm{Unpr}(J,R) = \text{graph of } \phi = \frac{R[T]}{ (T \alpha-\phi(\alpha): \; \alpha \in J) }

“And that ring is Gorenstein?”

“Yes, it is. It is the Kustin-Miller unprojection” I laugh, trying to distract her as the army of nuteaters are kidnapping her brother.

“Hm. But you wanted to tell me how to normalize the fundamental class…”

Nothing escapes her. Well, almost nothing, I think, seeing her sedated brother getting carried off.

“Get me a soylent tea.” I ask her, and flick a knife in the direction of those darned hazelnut fanatics, sending them scurrying away. Her brother is free. Safe. But for how long? How long will they live? I will not tell her now how Kustin-Miller unprojection is used usually, for the construction of interesting rings. For explicit birational geometry. She has to worry about other things blowing up.

A tear escapes my eye. A metaphorical one. They have taken our tearducts to make “New Coke”. Little Gretchen returns.

“Go on, Opa.”

She begs me. I don’t want to, but those eyes.

“Please”

Those darned eyes. “Consider a standard Gorenstein algebra

R = k[x_1, \dots , x_m] / I

where k is any field. Let’s say it is of Krull dimension p“. Krull, like the hero of the bunnywars. We would not be here without him. Good or bad.

“Let’s do the Artinian reduction” she rejoices. Our favorite game. “And how do we do that”

Her fingers draw diagrams into the muck as I slurp my human tea, nodding approvingly.

“We consider new variables z and a_{i,j} and

f_i = \sum_{j=1}^m a_{i,j} x_j

as well as

B = k [ z, a_{i,j}, x_1 , \dots , x_m ] / (I, f_1, \dots , f_p).

Then J = (z, x_1 , \dots , x_m) in B satisfies the assumptions for Kustin-Miller unprojection.”

She looks at me, incredibly bright. She knows what is coming, mouthing the same words in her mouth as I speak them.

“So there is \phi: J \to B as before and…” I remember a beautiful seminar talk by Leonid Monin that I had slept through (sorry Leonid). I catch myself. Another ghostly tear. My voice breaks.

“and we can choose h \in k [ z, a_{i,j}, x_1 , \dots , x_m ] so that \phi (z) = h.”

Her eyes light up. She knows it, I know it. Only little Max, dragging himself in, little bits of fur in his teeth, does not. He will outlive us all.

“And we normalize \mathrm{deg} A^{\mathrm{top}}\rightarrow k by setting

\mathrm{deg} ( \; h \; ) = 1.“

Now it gets temporally tricky. Because I don’t know whether I will know that I would have had found that out in the meantime. For now I only know that I will know that I had had known a few special cases. Feel free to correct my time travel grammar here.

“At least in a few cases. You see, it turns out that with this choice of \mathrm{deg} is the canonical one in the case of toric varieties: as a rational function depending on the a_{i,j}, it coincides with the degree map and”

She giggles with joy.

“we proposed that this is also the right one in the quasitoric case, for semigroup algebras.”

Her eyes widen. I am exhausted. It is late, the tea is almost empty. The razor tornadoes will come soon. But I cannot deny her energy.

“Why?” she asks that childish question.

“Because, for one, it gives beautiful answers in these cases; for instance, in the case that we are considering Gorenstein algebras that are locally semigroup algebras, this leads to a local definition, in that the degree map of a monomial is only dependent on those a_{i,j} that are in the poset ideal of the monomial.” I explain things I know I will know, but do not know yet and that I leave for the next paper. I explain how this leads to a Lefschetz property. But she is falling asleep. Her last mumbles are distant, silent, drowned out by the marauding koalas outside.

“So if people had known this, it would have…”

“saved the world?” I caress her hair as she drifts asleep, carefully combing the tigerlice out of her hair.

“Yes. Yes it would have.” I will sigh, and watch the world burn.

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