Tropical Algebraic Geometry - A tropical count of binodal cubic surfaces
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Geiger, Alheydis
Brandt, Madeline
In this talk I will give a short introduction to tropical algebraic geometry based on a tropical count of binodal cubic surfaces (joint work with Madeline Brandt). There are 280 binodal cubic surfaces through 17 points in general position. They can be counted using tropical geometry. After a brief introduction into tropical geometry we see how the dual subdivision of the Newton polytope allows us to count surfaces through points in Mikhalkin position via floor plans. We will see that we can recover 214 binodal surfaces with separated nodes in their tropicalizations and we will also have a short look at the complexes hiding the remaining 66 surfaces.
 Hello, my name is Alhidus Gargar. I am a first year PhD student at the University of Tuvingen. First, I want to thank the organizers for the invitation to speak here at the ICMS session on polyhedron methods in geometry and optimization. Today I will give you a brief introduction into tropical geometry and its applications in enumerative geometry. By giving you an insight into the intertropical counting by nodal cubic surfaces, this talk is based on joint work with Mellon Bund. Now we are going to see how tropical geometry helps solving enumerative questions. In enumerative geometry, we are interested in the number of geometric objects which satisfy certain conditions. Classical examples are we are looking at curves as our geometric objects and we let the curves satisfy point conditions. For example, we have a Grammar Fittian variant which is the number of degree D rational curves passing through 3D minus 1 point. This is counting complex curves but we can also be interested in the number of real curves and for this we have the Vajorjian variants and those are a signed count of real degree D rational curves. Tropical geometry provides useful tools to solve such enumerative questions. For example, there are a lot of successes using tropical geometry when investigating the enumeration of curves. The first and most important of these is the so-called correspondence theorem which was proven by Mikakin in 2003. The correspondence theorem tells us that if we count in the tropical world, we obtain the same number as if we are counting in the algebraic world. This means that all the tools tropical geometry provides can be used for the count without changing the outcome. Counting tropical geometry means that Counting tropical geometry has been possible to show that the Vajorjian variants which are a signed count of real curves are actually always positive. This is a big result and it has so far not been possible to prove this without using tropical. We also know that the Vajorjian variants grow logarithmically asymptotically like the Grammar Victor variant and this is also only known by tropical geometry. We see tropical geometry is a powerful tool when investigating the enumeration of curves. What is next? Of course, we can pass through higher dimensional varieties like surfaces. We want to use tropical geometry to count surfaces that satisfy quantifications. Before we go into that, let me explain what tropical geometry actually is. Tropical people is a combinatorial shadow of other variety geometry. That sounds very mystical. Let's see what this actually means. If we want to do tropical geometry, we have to fix a field with an onachemetic evaluation. Now we take a polynomial and it's algebraic hyper surface. These are objects you all know. Now we want to take these algebraic objects and push them into the tropical world. This pushing into the tropical world is called tropicalizing. What happens is that we pass over to the tropical semi-field. This works in the following way. The tropical semi-field has the operations tropical addition, tropical multiplication. Our usual addition and multiplication will pass over and become these tropical operations. The tropical addition is taking the maximum. Tropical multiplication is what we know as addition. Also when tropicalizing, we have to take our coefficients and take the negative evaluation in order to ensure that we are over the real numbers because the tropical semi-field as a set is the real numbers with negative infinity. What we get as the tropical polynomial of f is the maximum of all a minus evaluation ca plus a times x. As you see, this is a piecewise linear function. Now if we're looking at the hyper surface and want to tropicalize this, it turns out and this is the fundamental theorem of the tropical algebraic geometry that this hyper surface is the same as taking all those points in r to the n where the maximum of x is achieved at least twice. This is the same as just taking the valuation of all the points in the hyper surface. It's just the same as we do with the tropical polynomial. What we do is we pass over through the tropical semi-field, meaning taking the valuation and changing our operations accordingly. For the hyper surface, we just take all the valuation and it turns out it's the same as looking at the points where the maximum in the tropical polynomial is achieved twice. This might look complicated to you and you might ask yourself why are we doing this. We're doing this because we can encode all the information in the dual subdivision of the Newton's polytope of the polynomial. The Newton's polytope is the convex hull of those exponents of x where the coefficient is non-zero and the negative valuations induce a subdivision on this. This encodes all the information of our tropical hyper surface. That means that we have polyhedral geometry and combinatorial tools at our fingertips when we investigate these tropical objects. Let's look at an example to see how this works. I've brought you a tropical polynomial. The first thing we want to do is we want to find the Newton's polytope. For the Newton's polytope, we need to look at the exponents of x. We have zero zero here. We have one zero zero one zero two. Oh, two zero here. I forgot the two x and one one. Which means our Newton's polytope as the convex hull of these points will look like this. It's a triangle. The next thing was we have to induce the subdivision. How does this work? For each of the lattice points in our Newton's polytope, we assign the value of these coefficients. In our case, these are just the constant parts of each of the linear sections we're taking the maximum off. This will look like this. It's just the triangle assigned these values on each lattice point. What you now imagine is that you take each of the lattice points and you pull it out of the screen just the number of steps that the number tells you that you assigned to the vertex. There's three on top here. We'll be pulled out three steps. Two will be pulled out two steps. Zero will just stay on the screen. If you now look at this, take the convex hull you'll get, kind of roof. This roof has breaking lines. One linear part breaks into the next. If we project these back down onto the screen, you will obtain this subdivision. This is the dual subdivision. This object now contains all the information. If we just take our steps backwards, we'll end up with this tropical polynomial that we started with. We'll get exactly that one with no difference. What we can also obtain is the hypersurface in our case will be a curve by the concept of duality. You can just imagine with this maximum convention that we're having, we just draw the orthogonal rays in here, the edges, and end up with this. The heights that we assign to the lattice points determine the position of the vertices of the graph. Okay, great. Now that you have a small inside and tropical geometry, let's start with counting surfaces. Okay. If we count surfaces, we'll start with the easiest case, which is smooth surfaces. We look at the parameter space. The parameter space is P19 because P19 parameterizes all cubic surfaces in P3. Now we're interested in the number of smooth cubic surfaces passing through 19 points. That is the correct number of points. Okay. This is exactly one. As you should all know, it's a classical algebraic result. There is exactly one. How can I count this tropical? I need to fix my point conditions on my tropical surface, and I need to know the surface is smooth. Now, since everything is piecewise linear in tropical geometry, we need to understand what smoothness means tropical. Tropically smooth means that the dual subdivision of this surface or of this geometric object is a unimodular triangulation. Okay. What does that mean? Triangulation means that all the maximum polytopes that we have are simplices. The Newton polytope to a smooth cubic surface is a 3 by 3 tetrahedron. It's three-dimensional. All the sub-pollutopes in the subdivision need to be also tetrahedral because that is a three-simplex. It has to be unimodular, and unimodular means that as let us polytopes, they have to have minimum volume. For a tetrahedron, this means it has to have volume 106. What we have is we have a new polytope, which is this 3 by 3 tetrahedron. We get our point conditions, and we now have the smoothness. These are all the conditions, and these conditions actually determine the subdivisions uniquely. If we have our fixed points, and we know the subdivision has to be unimodular triangulation, we get a unique subdivision, and therefore a unique tropical surface. Okay. Fine. What if we have nodes? Classically, we know if we have nodes, we're looking at the set of singular cubic surfaces. This is a variety, and it is a codimension, one variety. Now we can simply use Bezu's theorem, which tells us that the number of singular cubic surfaces passing through 18 points is the degree of the set of singular cubic surfaces of this variety. This variety has degree 32, so we have 32 singular cubic surfaces passing through 18 points. Great. So what about the tropical side of this? How can I count this tropical? And for this, we first need to think about how a node in a tropical cubic surface looks like. As I told you, tropical everything is pizza-wise linear. So what is the concept of a node? And it is that we just say a surface is tropical delta nodal, even if it is the tropicalization of a delta nodal surface. So we're just saying it's delta nodal down in the tropical world, if there is an anorgan in the algebraic world which contains a node, then this would tropicalize down. Okay, so we know that if our tropical surface is smooth, the triangulation is unimodular. So if it's the tropicalization of a delta nodal surface, tropicalization cannot be unimodular triangulation. So there has to be something inside our subdivision which is not unimodular tetrahedron. And now we see here some polychip complexes. And these were polychip complexes that were found by Thomas Maverick, Hannah Maverick and Gjafge and Ischestin. And they show that if we have our point conditions in a special way, and these are points in the calcine position, and this is tropical still generic, and it's also the lifts are also algebraically generic, so this is all fine, it's all generic. But they are especially nice for counting tropical objects. Now if we have our points in this special but generic position, a single node will have to be encoded in the subdivision by one of these polychip complexes. And if we have multiple nodes, and they are separated, that means that one node arises from one of these complexes and the complexes don't interfere with each other in the subdivision, which means they're just intersecting in a unimodular phase. This all sounds theoretically rather complicated, so let's look at an example that might make it better to understand what we're doing. If we're counting one node of surfaces, we need to find a way how to put these polychip complexes, the pentatope or the bipyramid as you see here, or this weight 2 end configuration in our subdivision, such that the subdivision apart from this special complex is as unimodular as is possible. In our case, we just need to find all the ways how to put one single bipyramid inside our subdivision, or we need to find all the ways, and there are multiple ways, I'm not going to show them all to you, I just have a few pictures, and we need to find all the ways for the pentatope, it turns out for one node there is no pentatope, and we have to find all the ways for this weight 2 configuration, you see it here, this is an edge of length 2, so it will be an unbounded 2 cell of weight 2, ql2 in the surface, and there are also different ways to put this in. And if we enumerate all of these and count with a lifting on to 50, we'll end up with a 32 surfaces. So for one node surfaces, we can just count trabically with what we already know, and it works out nicely. Okay, so what about binodal cubic surfaces, that's the next step, right? So we're looking at the singular locus of the set of singular cubic surfaces, and this is the union of the binodal cubic surfaces and the caspital cubic surfaces, so there are varieties of codemandu2, so what we want to know is the degree of the binodal locus, or set differently, the number of binodal cubic surfaces passing through 17 points. And classically, this question was solved by Vintcensure in 2003, and he computed formulas for k-node degree D surfaces, here you see the formula for a binodal surface of degree D, so if you plug in D equal 3 for our cubic surfaces, we'll get the number and it is 280. So quite a lot bigger than 32. And now we want to see how we can count this trabically. And by Hannah Maakwijk, Thomas Maakwijk, Christensho and Jeff Gainich's team, it was proven in 2019 that if we're counting asymptotically, we can recover the leading coefficient of the number of delta-node degree D surfaces. The number of delta-node degree D surfaces is given by a polynomial, but all we know so far are the leading terms of this polynomial, and if we count trapically and asymptotically, we can recover this leading coefficient. And the concept, the methods, the tropical methods they developed are called tropical floor plans, and they make this combinatorial question of deciding how many ways are there to put the bi-primate inside our subdivision or the pentatube or the way to configuration. The tropical floor plans break this question down to something that's a little easier to handle. Now if you're counting the actual accurate number, and not asymptotically, but the accurate number, for binodal surfaces, so far there are only 9,214 complex cubic surfaces that can be counted like this. And these are the surfaces that contain separated singularities. That means the singularities come from the polytope complexes that I showed you before, the pentatube, bi-primate, and way to configuration, and from those two. So we have one node that corresponds to one of those complexes and the other one to another one. But what also can happen is, so you can see a picture, here you see bi-primate and way to configuration, and here a pentatube. But they can also be complexes that give rise to two nodes at the same time. So this one complex contains two nodes and I can't separate them out. For example, we have two nodes contained in the same two cell or on the same edge or something like this. And these are the complexes that we still don't understand. We don't know, these are the ones we found, and we don't know how to count them. We don't know do they contain two nodes and how do they contain these nodes. And this is what Madeline and I are still investigating. Okay, let's conclude. Please remember, what I first told you about tropical geometry, it combines polyhedron combinatorial methods and everything is piecewise linear. When we're counting surfaces using tropical geometry, we enumerate the subdivisions. And using this, we can completely recover the count of one-node cubic surfaces and also asymptotic linear, we obtain at the right number. However, if we're counting the accurate number, there still needs to be something done because so far we only understand 214 surfaces and remaining 66 surfaces with non-separated nodes still need to be investigated. Here's a little literature. If you're interested, the pictures I showed you were made using PolyMake. And this is a great software that you have or will hear about in another talk. All right, so thank you for your attention.
 Hello, my name is Alhidus Gargar.  I am a first year PhD student at the University of Tuvingen.  First, I want to thank the organizers for the invitation to speak here at the ICMS session  on polyhedron methods in geometry and optimization.  Today I will give you a brief introduction into tropical geometry and its applications  in enumerative geometry.  By giving you an insight into the intertropical counting by nodal cubic surfaces, this talk  is based on joint work with Mellon Bund.  Now we are going to see how tropical geometry helps solving enumerative questions.  In enumerative geometry, we are interested in the number of geometric objects which satisfy  certain conditions.  Classical examples are we are looking at curves as our geometric objects and we let the curves  satisfy point conditions.  For example, we have a Grammar Fittian variant which is the number of degree D rational curves  passing through 3D minus 1 point.  This is counting complex curves but we can also be interested in the number of real curves  and for this we have the Vajorjian variants and those are a signed count of real degree  D rational curves.  Tropical geometry provides useful tools to solve such enumerative questions.  For example, there are a lot of successes using tropical geometry when investigating  the enumeration of curves.  The first and most important of these is the so-called correspondence theorem which was  proven by Mikakin in 2003.  The correspondence theorem tells us that if we count in the tropical world, we obtain  the same number as if we are counting in the algebraic world.  This means that all the tools tropical geometry provides can be used for the count without  changing the outcome.  Counting tropical geometry means that  Counting tropical geometry has been possible to show that the Vajorjian variants which  are a signed count of real curves are actually always positive.  This is a big result and it has so far not been possible to prove this without using tropical.  We also know that the Vajorjian variants grow logarithmically asymptotically like the Grammar  Victor variant and this is also only known by tropical geometry.  We see tropical geometry is a powerful tool when investigating the enumeration of curves.  What is next?  Of course, we can pass through higher dimensional varieties like surfaces.  We want to use tropical geometry to count surfaces that satisfy quantifications.  Before we go into that, let me explain what tropical geometry actually is.  Tropical people is a combinatorial shadow of other variety geometry.  That sounds very mystical.  Let's see what this actually means.  If we want to do tropical geometry, we have to fix a field with an onachemetic evaluation.  Now we take a polynomial and it's algebraic hyper surface.  These are objects you all know.  Now we want to take these algebraic objects and push them into the tropical world.  This pushing into the tropical world is called tropicalizing.  What happens is that we pass over to the tropical semi-field.  This works in the following way.  The tropical semi-field has the operations tropical addition, tropical multiplication.  Our usual addition and multiplication will pass over and become these tropical operations.  The tropical addition is taking the maximum.  Tropical multiplication is what we know as addition.  Also when tropicalizing, we have to take our coefficients and take the negative evaluation  in order to ensure that we are over the real numbers because the tropical semi-field as  a set is the real numbers with negative infinity.  What we get as the tropical polynomial of f is the maximum of all a minus evaluation  ca plus a times x.  As you see, this is a piecewise linear function.  Now if we're looking at the hyper surface and want to tropicalize this, it turns out  and this is the fundamental theorem of the tropical algebraic geometry that this hyper  surface is the same as taking all those points in r to the n where the maximum of x is achieved  at least twice.  This is the same as just taking the valuation of all the points in the hyper surface.  It's just the same as we do with the tropical polynomial.  What we do is we pass over through the tropical semi-field, meaning taking the valuation and  changing our operations accordingly.  For the hyper surface, we just take all the valuation and it turns out it's the same as  looking at the points where the maximum in the tropical polynomial is achieved twice.  This might look complicated to you and you might ask yourself why are we doing this.  We're doing this because we can encode all the information in the dual subdivision of  the Newton's polytope of the polynomial.  The Newton's polytope is the convex hull of those exponents of x where the coefficient  is non-zero and the negative valuations induce a subdivision on this.  This encodes all the information of our tropical hyper surface.  That means that we have polyhedral geometry and combinatorial tools at our fingertips  when we investigate these tropical objects.  Let's look at an example to see how this works.  I've brought you a tropical polynomial.  The first thing we want to do is we want to find the Newton's polytope.  For the Newton's polytope, we need to look at the exponents of x.  We have zero zero here.  We have one zero zero one zero two.  Oh, two zero here.  I forgot the two x and one one.  Which means our Newton's polytope as the convex hull of these points will look like this.  It's a triangle.  The next thing was we have to induce the subdivision.  How does this work?  For each of the lattice points in our Newton's polytope, we assign the value of these coefficients.  In our case, these are just the constant parts of each of the linear sections we're taking  the maximum off.  This will look like this.  It's just the triangle assigned these values on each lattice point.  What you now imagine is that you take each of the lattice points and you pull it out  of the screen just the number of steps that the number tells you that you assigned to  the vertex.  There's three on top here.  We'll be pulled out three steps.  Two will be pulled out two steps.  Zero will just stay on the screen.  If you now look at this, take the convex hull you'll get, kind of roof.  This roof has breaking lines.  One linear part breaks into the next.  If we project these back down onto the screen, you will obtain this subdivision.  This is the dual subdivision.  This object now contains all the information.  If we just take our steps backwards, we'll end up with this tropical polynomial that  we started with.  We'll get exactly that one with no difference.  What we can also obtain is the hypersurface in our case will be a curve by the concept  of duality.  You can just imagine with this maximum convention that we're having, we just draw the orthogonal  rays in here, the edges, and end up with this.  The heights that we assign to the lattice points determine the position of the vertices  of the graph.  Okay, great.  Now that you have a small inside and tropical geometry, let's start with counting surfaces.  Okay.  If we count surfaces, we'll start with the easiest case, which is smooth surfaces.  We look at the parameter space.  The parameter space is P19 because P19 parameterizes all cubic surfaces in P3.  Now we're interested in the number of smooth cubic surfaces passing through 19 points.  That is the correct number of points.  Okay.  This is exactly one.  As you should all know, it's a classical algebraic result.  There is exactly one.  How can I count this tropical?  I need to fix my point conditions on my tropical surface, and I need to know the surface is  smooth.  Now, since everything is piecewise linear in tropical geometry, we need to understand  what smoothness means tropical.  Tropically smooth means that the dual subdivision of this surface or of this geometric object  is a unimodular triangulation.  Okay.  What does that mean?  Triangulation means that all the maximum polytopes that we have are simplices.  The Newton polytope to a smooth cubic surface is a 3 by 3 tetrahedron.  It's three-dimensional.  All the sub-pollutopes in the subdivision need to be also tetrahedral because that is  a three-simplex.  It has to be unimodular, and unimodular means that as let us polytopes, they have to have  minimum volume.  For a tetrahedron, this means it has to have volume 106.  What we have is we have a new polytope, which is this 3 by 3 tetrahedron.  We get our point conditions, and we now have the smoothness.  These are all the conditions, and these conditions actually determine the subdivisions uniquely.  If we have our fixed points, and we know the subdivision has to be unimodular triangulation,  we get a unique subdivision, and therefore a unique tropical surface.  Okay.  Fine.  What if we have nodes?  Classically, we know if we have nodes, we're looking at the set of singular cubic surfaces.  This is a variety, and it is a codimension, one variety.  Now we can simply use Bezu's theorem, which tells us that the number of singular cubic  surfaces passing through 18 points is the degree of the set of singular cubic surfaces  of this variety.  This variety has degree 32, so we have 32 singular cubic surfaces passing through 18 points.  Great.  So what about the tropical side of this?  How can I count this tropical?  And for this, we first need to think about how a node in a tropical cubic surface looks  like.  As I told you, tropical everything is pizza-wise linear.  So what is the concept of a node?  And it is that we just say a surface is tropical delta nodal, even if it is the tropicalization  of a delta nodal surface.  So we're just saying it's delta nodal down in the tropical world, if there is an anorgan  in the algebraic world which contains a node, then this would tropicalize down.  Okay, so we know that if our tropical surface is smooth, the triangulation is unimodular.  So if it's the tropicalization of a delta nodal surface, tropicalization cannot be  unimodular triangulation.  So there has to be something inside our subdivision which is not unimodular tetrahedron.  And now we see here some polychip complexes.  And these were polychip complexes that were found by Thomas Maverick, Hannah Maverick  and Gjafge and Ischestin.  And they show that if we have our point conditions in a special way, and these are points in the  calcine position, and this is tropical still generic, and it's also the lifts are also  algebraically generic, so this is all fine, it's all generic.  But they are especially nice for counting tropical objects.  Now if we have our points in this special but generic position, a single node will have  to be encoded in the subdivision by one of these polychip complexes.  And if we have multiple nodes, and they are separated, that means that one node arises  from one of these complexes and the complexes don't interfere with each other in the subdivision,  which means they're just intersecting in a unimodular phase.  This all sounds theoretically rather complicated, so let's look at an example that might make  it better to understand what we're doing.  If we're counting one node of surfaces, we need to find a way how to put these polychip  complexes, the pentatope or the bipyramid as you see here, or this weight 2 end configuration  in our subdivision, such that the subdivision apart from this special complex is as unimodular  as is possible.  In our case, we just need to find all the ways how to put one single bipyramid inside  our subdivision, or we need to find all the ways, and there are multiple ways, I'm not  going to show them all to you, I just have a few pictures, and we need to find all the  ways for the pentatope, it turns out for one node there is no pentatope, and we have to  find all the ways for this weight 2 configuration, you see it here, this is an edge of length  2, so it will be an unbounded 2 cell of weight 2, ql2 in the surface, and there are also  different ways to put this in.  And if we enumerate all of these and count with a lifting on to 50, we'll end up with  a 32 surfaces.  So for one node surfaces, we can just count trabically with what we already know, and  it works out nicely.  Okay, so what about binodal cubic surfaces, that's the next step, right?  So we're looking at the singular locus of the set of singular cubic surfaces, and this  is the union of the binodal cubic surfaces and the caspital cubic surfaces, so there  are varieties of codemandu2, so what we want to know is the degree of the binodal locus,  or set differently, the number of binodal cubic surfaces passing through 17 points.  And classically, this question was solved by Vintcensure in 2003, and he computed formulas  for k-node degree D surfaces, here you see the formula for a binodal surface of degree  D, so if you plug in D equal 3 for our cubic surfaces, we'll get the number and it is 280.  So quite a lot bigger than 32.  And now we want to see how we can count this trabically.  And by Hannah Maakwijk, Thomas Maakwijk, Christensho and Jeff Gainich's team, it was  proven in 2019 that if we're counting asymptotically, we can recover the leading coefficient of the  number of delta-node degree D surfaces.  The number of delta-node degree D surfaces is given by a polynomial, but all we know  so far are the leading terms of this polynomial, and if we count trapically and asymptotically,  we can recover this leading coefficient.  And the concept, the methods, the tropical methods they developed are called tropical  floor plans, and they make this combinatorial question of deciding how many ways are there  to put the bi-primate inside our subdivision or the pentatube or the way to configuration.  The tropical floor plans break this question down to something that's a little easier  to handle.  Now if you're counting the actual accurate number, and not asymptotically, but the accurate  number, for binodal surfaces, so far there are only 9,214 complex cubic surfaces that  can be counted like this.  And these are the surfaces that contain separated singularities.  That means the singularities come from the polytope complexes that I showed you before,  the pentatube, bi-primate, and way to configuration, and from those two.  So we have one node that corresponds to one of those complexes and the other one to another  one.  But what also can happen is, so you can see a picture, here you see bi-primate and way  to configuration, and here a pentatube.  But they can also be complexes that give rise to two nodes at the same time.  So this one complex contains two nodes and I can't separate them out.  For example, we have two nodes contained in the same two cell or on the same edge or something  like this.  And these are the complexes that we still don't understand.  We don't know, these are the ones we found, and we don't know how to count them.  We don't know do they contain two nodes and how do they contain these nodes.  And this is what Madeline and I are still investigating.  Okay, let's conclude.  Please remember, what I first told you about tropical geometry, it combines polyhedron  combinatorial methods and everything is piecewise linear.  When we're counting surfaces using tropical geometry, we enumerate the subdivisions.  And using this, we can completely recover the count of one-node cubic surfaces and also  asymptotic linear, we obtain at the right number.  However, if we're counting the accurate number, there still needs to be something done because  so far we only understand 214 surfaces and remaining 66 surfaces with non-separated nodes  still need to be investigated.  Here's a little literature.  If you're interested, the pictures I showed you were made using PolyMake.  And this is a great software that you have or will hear about in another talk.  All right, so thank you for your attention.
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